Test: Quadratic Equations- 2

Given

• x∗x+4∗x=−4

• x² + 4x + 4 = 0

To Find: value of x?

Approach

1. As we have a quadratic equation x²+4x+4=0

1. , we will solve this equation to find out the value of x

Working Out

x²+4x+4=0

⇒(x+2)²=0

⇒x=−2

Answer: B

Steps 1 & 2: Understand Question and Draw Inferences

Step 3: Analyze Statement 1 independently

Not sufficient to get a unique value of the ratio x/y.

Step 4: Analyze Statement 2 independently

Multiplying both sides of the inequality with  will not impact the sign of inequality:

From the Wavy Line Method:

Not sufficient to determine the exact value.

Step 5: Analyze Both Statements Together (if needed)

From Statement 1:

From Statement 2:

Both the values of   obtained from Statement 1 is less than −1/5

Therefore, both these values satisfy Statement 2

So, we are still not able to determine a unique value of the ratio x/y

Answer: Option E

Given

To Find: value of n?

Approach

1. We know that the product of roots m and n is equal to c/a

1. So, we can write m*n = 98, i.e. n = 98/m
2. So, for finding n, we need to find m

• We are given that m = (p+q)/2

1. We know that sum of roots of the quadratic equation is −b/a
2. We will use this relation to find the value of p + q and hence the value of m

Hence, the other root of the equation x² + bx + 98 = 0 is -7

Answer: A

Given: 

.

To find: Which of the given 3 statements must be true about z?

Approach:

1. Looking at the 3 statements, we realize that in order to validate them, we’ll need to know the value of z
2. We’re given an expression for z². Using that expression, we’ll find the value of z

Working Out:

• Finding the value of z
  • ​
  • Notice that the series of  within  within   is never-ending. So, we can write:

To make our calculations easy, let’s replace z² with another variable, say 'y'

• So, we get: 

• Now, squaring both sides:

• y² = 2 + y
• y² – y – 2 = 0
• (y-2)(y+1) = 0
• So, y = 2 or y = -1
  • Since y = z² and perfect squares are never negative, y cannot be -1
• So, y = z² = 2
• This means, either  z√2   or z-√2

• Checking the validity of the 3 Statements
  • Statement I: Two values are possible for z
  • • As we’ve found above, indeed only 2 values are possible for z
    • Therefore, this statement is true
• Statement II: 4 - z² = 2
  • As we calculated above, z² = 2
  • So, 4 – z² = 4 – 2 = 2
  • So, this statement too is true
• Statement III: z⁸ = 16
  • Since z² = 2, (z²)⁴ = 2⁴
  • So, z⁸ = 16
  • Therefore, this statement is true as well

• Looking at the answer choices, we see that the correct answer is Option D

Given         

• ax²+bx+c=0

• • a ≠ 0
  • s>0
  • Roots of the equation are (r, s)
  • r = s + 50% of s = 1.5s
• r * s = 150

To Find:  r + s?

Approach

1. For finding the values of r + s, we need to find the values of r and s
2. We are given two relations between r and s, which are:
      a.  r = 1.5s and
      b.  r * s = 150
3. As we have 2 equations and 2 variables, we can find the values of r and s and hence the value of r + s.

Working Out

1. Finding values of r and s

r=1.5s……..(1)

r*s = 150……..(2)

Substituting r = 1.5s in (2), we have

1.5s² = 150

s² = 100, i.e. s = 10 or -10

1. If s = 10, r = 15
        a. So, r + s = 10 + 15 = 25
2. If s = -10,
   1. However, we are given s >0
   2. Hence, we can reject this case.
3. So, the value of r + s = 25

Hence the sum of the roots of the equation ax²+bx+c=0

 is 25

Answer: D

Given

• Let the roots be p and q, where p, q are integers
• p + q = prime number less than 10
  • p + q = {2, 3, 5, 7}
• p*q = prime number less than 10
  • So, p*q = {2, 3, 5, 7}
• Since the product of integers p and q is positive, this means that p and q are either both positive or both negative
  • But since p + q is positive, this means that p and q cannot both be negative.
  • Therefore p, q > 0

To Find:  Value of |p – q|?

• Note: we wrote the expression for difference between the roots as |p-q| and not simply p – q, because at this point, we do not know which of the 2 roots is bigger. By asking about the difference between the 2 roots, the question is asking us to find the value of (Bigger Root – Smaller Root)

Approach

1. To find the value of p – q, we need to find the value of p and q.
2. We know that pq = {2, 3, 5, 7}, i.e. product of two positive integers is prime, Now, the only possible way in which a prime number can be expressed as a product of two positive integers is (Prime number * 1)
3. Using the above information, we can find the possible values of (p, q) = (Prime number, 1), irrespective of the order (that is, irrespective of whether p = Prime Number and q = 1 or vice-versa).
4. Finally, we’ll eliminate those values of (p,q) for which p + q is not prime

Working Out

1. As pq = {2, 3, 5, 7}, possible values of (p, q) can be:

1. (p, q) = (2, 1) or (3, 1) or (5, 1) or (7, 1)

2. Checking if the sum p + q is prime for these possible values of (p,q):

1. If (p, q) = (2, 1), p + q = 3, which is prime
2. If (p, q) = (3, 1), p + q = 4, which is not prime
3. If (p, q) = (5, 1), p + q = 6, which is not prime
4. If (p, q) = (7, 1), p + q = 8, which is not prime

3. Hence, the only possible case where both pq and p + q are prime is when (p, q) = (2, 1), irrespective of the order.

4. So, p – q = 2 – 1 = 1

Answer: A

Given

To Find: Values of (a, b, c)

Approach

• For finding the values of (a, b, c), we would first need to find the value of x₁ , x₂
  • We will use the relation    
  • • to find out the values of x₁ , x₂
    • Also, we will keep in mind the constraint that x₁ , x₂ are integers

  • Now, we know that  

• We will use the above relation to find out the possible values of (a, b, c)

As x₁ , x₂ are integers, the possible cases for (x₁ , x₂₎ is either  (3,1) or (-3,-1)

•  Using (1), (2) and (3), the values of (a, b, c) can be of the form ( a, 4a, 3a) or (a, -4a, 3a)

• Among the options,
  • Option-I (-1, 4, -3) is of the form (a, -4a, 3a) and
  • Option- II (1, 4, 3) is of the form (a, 4a, 3a)

Hence, options I and II can be the value of ordered set (a, b, c).

Answer: D

Steps 1 & 2: Understand Question and Draw Inferences

• x, y are integers and xy < 0
  • x ≠ 0 and y ≠ 0
  • x and y are of opposite signs
• x² – y² = 0
  • (x+y) (x-y) = 0
  • Either x = y or x = -y
  • As x and y are of opposite signs, the case of x = y can be rejected.
  • Therefore, x = -y

To Find: Value of y

Step 3: Analyze Statement 1 independently

• As x is an integer, the case of x=−3/2 is rejected.
• Therefore x = 12
• Thus y = -x = -12

Sufficient to answer.

Step 4: Analyze Statement 2 independently

(2) y² = (-1)ⁿ (12y), where n is a positive integer not divisible by 2.

Step 5: Analyze Both Statements Together (if needed)

As we have a unique answer from steps 3 and 4, this step is not required.

Answer: D

Given

• ax²+bx+c=0

•  has two roots p and q a ≠ 0, 0 < y ≤ 1

To Find 

• Roots of ayx²+byx+cy=0

Approach

1. To find the roots of the equation ayx²+byx+cy=0 , we will do the following steps:
   1. Take y common and the get the equation in terms of ax2+bx+c=0
   2. As we know the roots of ax²+bx+c=0, we will use this knowledge to find the roots of the equation ayx²+byx+cy=0

•  

  Working Out

  Solving the equation ayx²+byx+cy=0

• ayx²+byx+cy=0
• y(ax²+bx+c)=0

So we have either y = 0 or ax²+bx+c=0

Since we are given that y > 0, y ≠ 0.

Hence ax²+bx+c=0

.As the roots of the equation ax2+bx+c=0  are p and q, the roots of the equation ayx²+byx+cy=0  will also be p and q.

Answer: A

Given

• Three quadratic equations:
  • x² – 16x + 55 = 0
  • 2x² – 4x - 70 = 0
  • x (x +7) = 44

To Find: Which pair of equations has at least one common root?

Approach

1. As we need to find the common roots among the equation, we will solve each quadratic equation to find their roots and then find the pair of equations that have at least one common root.

Working Out

So, the roots of Equations I, II and III are (5, 11), (-5, 7) and (-11, 4) respectively.

Hence, none of the equations have even one root in common.

Answer: E

Given:

Approach:

1. To answer the question, we need to know the possible values of x.
2. So, we’ll first solve Equation 1, then Equation 2 and then find the common roots of both these equations. These will be the possible values of x
3. Then, we’ll evaluate the 3 options.

Working Out:

• Solving Equation I
  •  2x² = 24 -x⁴  
  • We can factorize this equation as it is, but if powers like x⁴ intimidate you, then it’s better to substitute x² with a new variable, say z. That ways, the equation will look simpler:  2z = 24 – z²
  • Rearranging the terms: z² + 2z – 24 = 0
  • z² + 6z – 4z – 24 = 0
  • z(z+6) – 4(z+6) = 0
  • (z+6)(z-4) = 0
  • This means, z = -6 or 4
  • But z = x². So, x² = -6 or 4
    • Perfect square x² cannot be negative. Therefore, we can reject the negative root.
  • So, x² = 4
  • Therefore, x = +2 or -2
  • Thus, Equation I has 2 roots: 2 and -2
• Solving Equation II
  • 
  • This equation looks difficult primarily because it involves a square root term, . So, for the ease of our calculations, let’s substitute =y
  • So, the equation becomes: 
  • Multiplying both sides with 2y, we get: 2y² – 6 = y
  • Rearranging the terms: 2y² – y – 6 = 0
  • ​​
  • This means, y = -3/2 or y = 2
  • Since y =  we can write: 
  • 
  • However, since the square root of a number is always positive, the value of   cannot be -3/2 .
  • If you’ve any doubt about this, think: When you write z² = 16 and then take the square root on both sides, you write:  Either z = √16 or z = -√16. Note here that, in the ‘or’ case, you put the minus sign outside the value of √16. This minus sign makes z negative (we get z = -4 in the ‘or’ case) but √16 is always positive (equal to 4).
  • Similarly, if we have z² = x + 6,  then on taking the square root on both sides, we will write: Either  or
  • z= -  So, z may be positive or negative but  will always be positive.
  • So, =2
  • Upon squaring both sides of this equation, we get: x + 6 = 4
  • Therefore, x = -2
  • Thus, we get a single value of x from Equation II
• Finding the possible values of x
  • From Equation I: x = 2 or -2
  • From Equation II: x = -2
  • So, the value of x that satisfies both equations: x = -2
  • Thus, only 1 value of x is possible: {-2}
  • Evaluating the 3 options
    • Out of the 3 given options, x can only be equal to -2

  • Looking at the answer choices, we see that the correct answer is Option B

Steps 1 & 2: Understand Question and Draw Inferences

Given: x < 0

To find:  x = ?  

Step 3: Analyze Statement 1 independently

(1) 16x² – 16x – 5 = 0

 x²–x−5/16=0

• 2 values of x (roots) will be obtained from this quadratic equation
  • Product of these 2 values  = −5/16

• • • Since the product is negative, one root is positive and the other is negative

• The positive root will be rejected (given: x < 0)

Thus, a unique value of x will be obtained from Statement 1

Statement 1 is sufficient to answer the question.

Step 4: Analyze Statement 2 independently

x= -5/4 We can reject this value as we know that x < 0

Thus, we get 3 possible negative values of x from Statement 2:

Not sufficient to find a unique value of x

Step 5: Analyze Both Statements Together (if needed)

Since we get a unique answer in Step 3, this step is not required

Answer: Option A

Given: Let the total money be y

• First Investment:
  • ​
  • • Rate of interest = 2r% p.a. compounded every 6 months = r% per 6-months
  • Time = 1 year
• Second Investment:
  • ​
  • • Rate of interest = r% p.a.
    • Time = 1 year
• Total Interest earned in 1 year 

To find:  The value of r

Approach

1.  Total interest earned in 1 year = (Interest earned from 1^st investment) + (Interest earned from 2^nd investment)

i.  In the given time frame of 1 year, the 1^st investment will pay interest twice (since this investment pays interest every 6 months). So, the formula for compound interest will be applicable for the 1^st investment

ii.  The 2^nd investment pays interest after 1 year. Since the given time frame is also 1 year, this investment will yield simple interest

2.  The only unknown in the above equation will be r. So, using this equation, we can find the value of r

Working Out

• Amount of the first investment after 1 year 
• Compound Interest earned from 1^st investment 

1. Calculating the interest earned from the 2^nd investment:
   1. ​Simple interest earned from 2^nd investment =   
   2. So, Total interest earned in 1 year = 
      1. 
      2. Multiplying both sides of the above equation with 9/y :

• Rejecting the negative value since the money is given to have grown.

​

Looking at the answer choices, we see that Option C is correct

(Note: You could also have solved this question by framing the first equation in terms of the amount that each investment grows to, as under:

(Total Amount after 1 year) = (Amount that the 1^st investment grows to) + (Amount that the 2^nd investment grows to) 

This equation leads to a similar calculation and the same result as in the solution above /End of Note)

To Find: Number of negative values of z

Approach:

• Solve the given equation
  • Note that the denominator is z - 1.
  • To remove the denominator, we will need to multiply each term of this equation with z - 1.
  • This will be time consuming (because we will have to multiply each term with z as well as with -1. Example:   and is likely to result in a tedious calculation.
  • To simplify this process, we will substitute (z - 1) as x 
• Find the value of z
• Count the number of negative values of z

Substitute z - 1 = x  

Therefore z = x + 1 ; z - 2 =  x -1

Hence

This is a cubic equation (involves x³) and you may feel that you do not know how to solve a cubic equation.

However, before giving up, think about how you solve a quadratic equation? By rewriting the given equation into its factors.

Let’s try if the cubic equation above can be similarly written into factors. We’ll find that the middle term, -21x, can be broken down as under:

​

That is 

• z - 1 = 1 or 4 or -5
• z = 2 or 5 or -4
• Thus, we get three values of z. However, the question asks specifically about the number of negative values of z.
• Among the 3 possible values of z, we see that only one value of z is negative.

Correct Answer: Option B

Given

•   to an infinite terms
  • As the value of x is equal to the square root of something, x ≥ 0

To Find:

The value of x

Approach

x=  to an infinite terms.

• Since the expression under the root extends infinitely, the expression under the root will be equal to x itself.
• So,  0………….(1)
• By solving this equation and applying the constraint that x ≥0, we will find the value of x

Working Out

a.

b. Squaring both sides, we have 

As we know that x≥ 0, x = 4

   Answer: D

Alternate solution

• Once we have deduced that x will be positive the options -3 and -4 are eliminated
• We know √9 is 3. So the value of √(12+ ……) will  be more than 3. Now we are left with 4 and 12 in the options

• Now √(12 + …..) cannot give us 12..
• So the answer has be 4.

Steps 1 & 2: Understand Question and Draw Inferences

• ax²+bx+c=0

•  where a*b*c ≠ 0
  • This means, a ≠ 0, b ≠ 0 and c ≠ 0
• p and q are roots of the equation

To Find: pq > 0?

• Is pq > 0?
  • pq > 0, if p and q are of the same signs
  • pq < 0, if p and q are of the opposite signs
• Also, Product of roots = pq = c/a

• So, the question becomes: Is c/a>0?

• c/a>0?  , if c and a are of the same signs

• c/a<0  , if c and a are of the opposite signs

So, we need to find the signs of either (p and q) of (c and a).

Step 3: Analyze Statement (1) independently

(1) |p + q| = |p| + |q|

We know, pq = c/a , where a ≠ 0 and c ≠ 0, So p ≠ 0 and q ≠ 0

Now, we know that |x| = x if x ≥ 0 and |x| = -x if x < 0.Since, we have |p| and |q|, following cases are possible:

1. p > 0 and q >0
   • As p > 0 and q > 0, we have |p| = p and |q| = q
   • So, p + q > 0
   • Hence, we can write |p+q| = p +q and |p| +|q| = p + q.
   • So, |p+q| = |p| + |q|
   • Thus, the equation in statement-1 holds true for p > 0 and q > 0
   • In this case pq > 0
2. p > 0 and q < 0
   • As p > 0, we have |p| = p and as q < 0, we have |q| = -q
   • Now, |p+q| = p + q, if p + q > 0 or –(p+q), if p + q < 0
   • Also, |p| +|q| = p - q
   • So, for both the values of |p+q|, we have  |p + q| ≠ |p| + |q|
   • Thus, the equation in statement-1 does not hold true for p >0 and q < 0
3. p < 0 and q >0
   • As p < 0, we have |p| = -p and as q >0, we have |q| = q
   • Now, |p+q| = p + q, if p +q ≥ 0 or –(p+q) if p + q < 0
   • Also,  |p| + |q| = -p +q
   • So, for both the values of |p+q|, we have |p + q| ≠ |p| + |q|
   • Thus, the equation in statement-1 does not hold true for p < 0 and q >0
4. p < 0 and q < 0
   • As p < 0, we have |p| = -p and as q < 0, we have |q| = -q
   • Now, |p+q| = - (p+q) because p + q < 0
   • Also,  |p| + |q| = -p – q = - (p+q)
   • So, |p+q| = |p| + |q|
   • Thus, the equation in statement-1 holds true for p < 0 and q < 0
   • In this case pq > 0

So, |p+q| = |p| +|q| is possible only when p, q > 0 or p,q < 0

So in both the cases pq will be greater than 0.

Hence Sufficient to answer

Step 4: Analyze Statement (2) independently

(2) ac > 0

Tells us that a and c have the same signs. Thus c/a>0

 and hence pq > 0.

Sufficient to answer

Step 5: Analyze Both Statements Together (if needed)

As we have a unique answer from step 3 and step 4, this step is not required.

Answer: D

Learning

• |a+b| = |a| + |b| if a and b have same sign
• |a+b| < |a| + |b| if a and b have different sign

Alternatively,

• |a-b| = |a| - |b| if a and b have same sign
• |a-b| > |a| -|b| if a and b have different sign

Steps 1 & 2: Understand Question and Draw Inferences

Given:

Let the length and breadth of rectangle be L and B respectively

• LB = 28…..............................................(1)

To find:  The value of 2(L+B)

• To find perimeter, we need to know the value of L and B.

Step 3: Analyze Statement 1 independently

(1) If the length of the rectangle is increased by 10 centimeter and the breadth is decreased by 5 centimeter, the perimeter of the rectangle is eight times the original length of the rectangle.

• New L = L  +10
• New B = B - 5

So,

• New Perimeter = 2(L+10 + B-5)

• 2(L+10 + B-5) = 8L

• 2L + 2B + 10 = 8L
• 6L – 2B – 10 = 0
• 3L – B – 5 = 0
• B = 3L - 5 . . . . . . . . . . . . . . .. . . . .(2)

Substituting (2) in (1):

• 2 values of L (roots) will be obtained from this quadratic equation

Comparing this with the standard quadratic form : ax² + bx + c = 0, we get : a = 1 ; b = -5/3 ; c = -28/3

• Product of these 2 values  = (c/a)  −283

• • Since the product is negative, one root is positive and the other is negative

• The negative root will be rejected
  • L, being the length of a rectangle, cannot be negative
• Thus, a unique value of L is obtained
  • Using Equation (2), a unique value of B is also obtained

Hence, Statement 1 alone is sufficient.

Step 4: Analyze Statement 2 independently

• If the length of the rectangle is increased by 350% and the breadth is increased to 350% of the original breadth, the perimeter of the rectangle is 63 centimeters more than the original perimeter of the rectangle

• New L = 4.5L
• New B = 3.5B

So,

Substituting (3) in  (1):

• 2 values of L (roots) will be obtained from this quadratic equation

Comparing this with the standard quadratic form : ax² + bx + c = 0, we get :

a = 1 ; b = - 9 ; c = 20

• Product of these 2 values  = (c/a) =  20
  • Since the product is positive, the two roots are either both positive or both negative
• Sum of roots = (-b/a) =  -(-9) = 9
  • Since the sum is positive, it means both roots are positive
• Thus, St. 2 leads to 2 values of L
  • From Equation (3), 2 values of B will be obtained
  • So, 2 values of Perimeter will be obtained.

St. 2 is not sufficient to obtain a unique value of the perimeter.

Step 5: Analyze Both Statements Together (if needed)

Since we get a unique answer in Step 3, this step is not required

Answer: Option A

Steps 1 & 2: Understand Question and Draw Inferences

• x²+bx+c=0

• Let the distance of the roots of the equation from 5 be d.
  • So, the roots will be 5 +d and 5 – d
    • Since the 2 roots are distinct, d ≠ 0
  • The sum of the roots = (5+d) + (5-d) = 10
    • So, −b/a=10
    • In the given quadratic equation, a, which denotes the coefficient of x², is 1.
    • So b = -10

To Find: As we have assumed the roots of the equation to be (5 +d) and (5 –d), to find the roots, we need to find the value of d

Step 3: Analyze Statement 1 independently

(1) The product of the roots of the equation x² + bx + c = 0 is 21

• Product of roots = c/a=21

• Since a = 1 in this equation, we can infer that c = 21
• We need to find the roots of x²−10x+21=0

• Since, we know the quadratic equation, we can find the roots of the equation.

Sufficient to answer

Alternate Method:  

• Product of roots = (5+d) (5-d) = 21
• 5²−d²=21

• d² = 25 - 21 = 4
  d = 2 or -2
• For d = 2
  • Roots = 5+2 = 7 and 5-2 = 3
• For d = -2
  • Roots = 5-2 = 3 and 5+2 = 7

So, the roots of the equation are 3 and 7

Sufficient to answer

Step 4: Analyze Statement 2 independently

(2) x – 7 is a factor of the expression 7x² + 7bx + 7c

• x- 7 is a factor of the expression 7(x²+bx+c)

Since 7 is a constant

• (x-7) is a factor of the expression x2+bx+c

Therefore,

• x = 7 is one of the roots of the equation x2+bx+c=0

That is 

• 5 + d = 7,
  • So, d = 2 and hence the other root is 5 – d = 3. OR
• 5 – d = 7
  • So, d = -2 and hence the other root is 5 +(-2) = 3

In both the cases, we get the same values of the roots.

Sufficient to answer

Step 5: Analyze Both Statements Together (if needed)

As we have a unique answer from steps 3 and 4, this step is not required

Answer: D

Step 1 & Step 2: Understanding the Question statement and Drawing Inferences

Given Info:

• x²+4x+p=13

• Rewriting the above quadratic equation in standard form ax²+bx+c=0

• Subtracting 13 from both sides, we have

⇒ x²+4x+p−13=13−13

⇒ x²+4x+p−13=0

To find:

• We need to find the product of the quadratic equation → x²+4x+p−13=0

• To find the product of the roots: 

1.  We need to know the roots or 
2. Products of roots of the quadratic equation ax²+bx+c=0 , is given by  c/a

• So product of roots of the above quadratic equation will be (p−13)/1 , as →c=p-13 and a=1
• Now since we do not know the value of p-13, we will not be able to determine the product of roots of the quadratic equation.
• Thus we need to analyse the given statements further to determine the value of p, to be able to calculate the product of roots of the quadratic equation.

Step 3: Analyze statement 1 independently
 

Statement 1:

• -2 is one of the roots of the quadratic equation
• So, -2 will satisfy the given quadratic equation → x²+4x+p−13=0

⇒ (-2)² + 4(-2) + p - 13 = 0

⇒ 4 - 8 + p - 13 = 0

⇒ p = 17

• Now since we know the value of p, we will be able to find the value of p-13 and thus will be able to calculate the value of product of quadratic equation.
• Hence statement 1 is sufficient to answer the question.

Step 4: Analyze statement 2 independently
 

Statement 2:

• Quadratic equation has equal roots

• For equal roots, we will use the relation of sum of roots of the quadratic equation to determine the value of the equal root.
• Sum of roots of the quadratic equation ax²+bx+c=0 , is −b/a
• So sum of roots of the quadratic equation → x²+4x+p−13=0  will be −4/1  , where b=4 and a=1
• Now since both roots are equal and the sum of the roots is coming as -4, both roots will thus be equal to – 2 each.
• Now, the equal root → -2, will satisfy the given quadratic equation x²+4x+p−13=0

⇒(-2)² + 4(-2) + p - 13 = 0

⇒ p=17

• Now since we know the value of p, we will be able to find the value of p-13 and thus will be able to calculate the value of product of quadratic equation.
• Hence statement 2 is sufficient to answer the question

Step 5: Analyze the two statements together

• Since from statement 1 and statement 2, we are able to arrive at a unique answer, combining and analysing statements together is not required.

Hence the correct answer is option D

In quadratic equations of the form ax² + bx + c = 0. - b/a  represents the sum of the roots of the quadratic equation and c/a represents the product of the roots of the quadratic equation.

In the equation given a = 1, b = b and c = 72
So, the product of roots of the quadratic equation = 72/1 = 72
And the sum of roots of this quadratic equation = -b/1 = -b
We have been asked to find the number of values that 'b' can take.
If we list all possible combinations for the roots of the quadratic equation, we can find out the number of values the sum of the roots of the quadratic equation can take.
Consequently, we will be able to find the number of values that 'b' can take.

The question states that the roots are integers.
If the roots are r₁ and r₂, then r₁ * r₂ = 72, where both r₁ and r₂ are integers.
Possible combinations of integers whose product equal 72 are : (1, 72), (2, 36), (3, 24), (4, 18), (6, 12) and (8, 9) where both r₁ and r₂ are positive. 6 combinations.

For each of these combinations, both r₁ and r₂ could be negative and their product will still be 72.
i.e., r₁ and r₂ can take the following values too : (-1, -72), (-2, -36), (-3, -24), (-4, -18), (-6, -12) and (-8, -9). 6 combinations.

Therefore, 12 combinations are possible where the product of r₁ and r₂ is 72.
Hence, 'b' will take 12 possible values.