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All questions of Quadratic Equations for ACT Exam

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If one of the root of a quadratic equation with rational coefficients is rational, then other root must be
  • a)
    Imaginary
  • b)
    Irrational
  • c)
    Rational
  • d)
    None of these
Correct answer is option 'C'. Can you explain this answer?

Raghav Bansal answered
Also, αβ = r/p, which is also rational. α + β = (a+√b) + (a-√b) = 2a, a rational number and, αβ = (a+√b)(a-√b) = a² - b, a rational number. So, the other root of a quadratic equation having the one root as (a+√b) is (a-√b), where a and b are rational numbers.

If 2x + 5 > 2 + 3x and 2x - 3 ≤ 4x - 5, then x can take which of the following values?
  • a)
    -2
  • b)
    2
  • c)
    4
  • d)
    -4
Correct answer is option 'B'. Can you explain this answer?

Samuel Butler answered
Given Inequalities:
2x + 5 > 2 + 3x
2x - 3 ≤ 4x - 5

Solving the first inequality:
2x + 5 > 2 + 3x
5 - 2 > 3x - 2x
3 > x

Solving the second inequality:
2x - 3 ≤ 4x - 5
-3 + 5 ≤ 4x - 2x
2 ≤ 2x
1 ≤ x

Combining the solutions:
Since x must satisfy both inequalities, x must be greater than 1 (from the second inequality) and less than 3 (from the first inequality). Therefore, x can take the value of 2.

Correct Answer:
Therefore, option B) 2 is the correct value for x based on the given inequalities.

 so the least integral value of n is
  • a)
    3
  • b)
    -3
  • c)
    -4
  • d)
    4
Correct answer is option 'D'. Can you explain this answer?

Lavanya Menon answered
{(1 + i)/(1 - i)}n = 1
multiply (1 + i) numerator as well as denominator .
{(1 + i)(1 + i)/(1 - i)(1 + i)}n = 1
{(1 + i)²/(1² - (i)²)}n = 1
{(1 + i² +2i)/2 }n = 1
{(2i)/2}n = 1
{i}n = 1
we know, i4n = 1 where , n is an integer.
so, n = 4n where n is an integers
e.g n = 4 { because least positive integer 1 }
hence, n = 4

Practice Quiz or MCQ (Multiple Choice Questions) with solutions are available for Practice, which would help you prepare for "Quadratic Equations" under Quantitative Aptitude. You can practice these practice quizzes as per your speed and improvise the topic. The same topic is covered under various competitive examinations like - CAT, GMAT, Bank PO, SSC and other competitive examinations.
 
Q. Find the roots of the quadratic equation: x2 + 2x - 15 = 0?
  • a)
    -5, 3
  • b)
    3, 5
  • c)
    -3, 5
  • d)
     -3, -5
  • e)
    5, 2
Correct answer is option 'A'. Can you explain this answer?

Shweta Talwar answered
1.multiply 1st term and last term of the eqn: 1*15 =15
2.now, find factors of 15 that sum of those two factors is the middle term of the eqn.
5 and -3 : when multiplied ans is: -15
and when added ans is: +2
3.change the sign of the factors i.e now factors are -5 and 3
therefore ans is: -5 , 3 (option A)

Calculate the least whole number, which when subtracted from both the terms of the ratio 5 : 6 gives a ratio less than 17 : 22.
  • a)
    5
  • b)
    3
  • c)
    2
  • d)
    4
Correct answer is option 'C'. Can you explain this answer?

Ayesha Joshi answered
Given:
Initial ratio = 5 ∶ 6
Final ratio should be less than 17 ∶ 22
Calculation:
Let the least whole number that is needed to be subtracted be a.
According to the question,
(5 - a)/(6 - a) < 17/22
⇒ 5 × 22 - 22a < 17 × 6 - 17a 
⇒ 110 - 22a < 102 - 17a 
⇒ 110 - 102 < - 17a + 22a 
⇒ 8 < 5a 
⇒ 8/5 = 1.6 < a 
∴ The least whole number must be 2.

Solve the quadratic equation x2 – ix + 6 = 0
  • a)
    1+2i, 1-2i
  • b)
    -2, 3
  • c)
    -2i, 3i
  • d)
    2, -3i
Correct answer is option 'C'. Can you explain this answer?

Nandini Iyer answered
x2 - ix + 6 = 0
x2 - 3ix + 2ix - 6i2 = 0    { i2 = -1}
x(x-3i) + 2i(x-3i) = 0
(x+2i) (x-3i) = 0
x = -2i, 3i 

ax2 + bx + c > 0 is __________
  • a)
    double inequality
  • b)
    quadratic inequality
  • c)
    numerical inequality
  • d)
    linear inequality
Correct answer is option 'B'. Can you explain this answer?

Ayesha Joshi answered
  • Since it has highest power of x ‘2’ and has inequality sign so, it is called quadratic inequality.
  • It is not numerical inequality as it does not have numbers on both sides of inequality.
  • It does not have two inequality signs so it is not double inequality.

In the Maths Olympiad of 2020 at Animal Planet, two representatives from the donkey’s side, while solving a quadratic equation, committed the following mistakes: (i) One of them made a mistake in the constant term and got the roots as 5 and 9. (ii) Another one committed an error in the coefficient of x and he got the roots as 12 and 4.But in the meantime, they realised that they are wrong and they managed to get it right jointly. Find the quadratic equation.
  • a)
    x2 + 4x + 14 = 0
  • b)
    2x2 + lx - 24 = 0
  • c)
    x2 - 14x + 48 = 0
  • d)
    3x2 - 17x + 52 = 0
Correct answer is option 'C'. Can you explain this answer?

Dhruv Mehra answered
STUDENT 1:

Roots: 5,9.

Hence, x= 5..or x=9

or, x-5 = 0 or x - 9 = 0

or, (x - 5)(x - 9) = 0

or, x2 - 14x + 45 = 0

STUDENT 2:

Roots: 12,4

Hence, x= 12..or x=4

or, x - 12 = 0 or x - 4 = 0

or, (x - 12)(x - 4) = 0

or, x2 - 14x + 48 = 0

ax + b > 0 is ___________
  • a)
    double inequality
  • b)
    quadratic inequality
  • c)
    numerical inequality
  • d)
    linear inequality
Correct answer is option 'D'. Can you explain this answer?

Ayesha Joshi answered
  • Since it has highest power of x ‘1’ and has inequality sign so, it is called linear inequality.
  • It is not numerical inequality as it does not have numbers on both sides of inequality.
  • It does not have two inequality signs so it is not double inequality.

The solution of the quadratic equation: 2x2 + 3ix + 2 = 0
  • a)
  • b)
  • c)
  • d)
Correct answer is option 'C'. Can you explain this answer?

Rajesh Gupta answered
2x2 + 3ix + 2 = 0
Using quadratic equation;
we know, x = (-b ± √b2 - 4ac)/2a
x =  [-3i ± √(3i)2 - 4x2x2]/2x2
= -3i ± √-25/4
= i(-3±5)/4
x = i/2, -2i

If x + 2y ≤ 3, x > 0 and y > 0, then one of the solution is
  • a)
    x = -1, y = 2
  • b)
    x = 2, y = 1
  • c)
    x = 1, y = 1
  • d)
    x = 0, y = 0
Correct answer is option 'C'. Can you explain this answer?

Orion Classes answered
Given 
x + 2y ≤ 3
x > 0 and y > 0
Calculation 
We need to satisfy the equation x + 2y ≤ 3 from the options 
Option: 1  x = -1 and y = 2 
This will be incorrect as we have x and y > 0 
In 1st option x is less than 0, so we can't take this 
Option: 2  x = 2, y = 1
2 + 2 ≤ 3 , which is incorrect.
Option: 3  x = 1, y = 1 
1 + 2 ≤  3 
3 ≤ 3, which is correct.
∴ The correct answer is x = 1, y = 1 

The roots of the equation 3x2 - 12x + 10 = 0 are?
  • a)
    rational and unequal
  • b)
     complex
  • c)
    real and equal
  • d)
    irrational and unequal
  • e)
    rational and equal
Correct answer is option 'D'. Can you explain this answer?

Gowri Chakraborty answered
The discriminant of the quadratic equation is (-12)2 - 4(3)(10) i.e., 24. As this is positive but not a perfect square, the roots are irrational and unequal.

The expression x2 + kx + 9 becomes positive for what values of k (given that x is real)?
  • a)
    k < 6
  • b)
    k > 6
  • c)
    |K|<6
  • d)
    |k|< 6
Correct answer is option 'C'. Can you explain this answer?

Om Desai answered
Method to Solve :

If the roots are equal(double root) it means that discriminant of quadratic equation b^2-4ac=0
general form of quadratic equation is ax^2+bx+c=0
in this case a=1 b=k and c=9
b^2-4ac=0 then:

k^2-36=0
(k-6)(k+6)=0
k=6 or k=-6
For k=6 or k= -6 given equation has real and equal roots

Find the minimum value of the expression (p +1/p); p > 0.
  • a)
    1
  • b)
    0
  • c)
    2
  • d)
    Depends upon the value of p
Correct answer is option 'C'. Can you explain this answer?

Pallabi Deshpande answered
It should be p. 
Let's try plugging in some values for p. 
First let's take p=0.1 -> 0.1+1/0.1 = 0.1+10/1 = 10.1 
Now let's take p=1 -> 1+1/1 = 2 (smaller Wink ) 
Now let's take p=2 -> 2+2/1 = 4 (bigger again) 

Therefore we know that the values will decrease if you plug in a number between ]0;1[, that the value will be minimum at 1 and later increase again.

If P and Q are the roots of f(x) = x2 - 14x + 45, then find the value of (1/P +1/Q)
  • a)
    45/14
  • b)
    14/45
  • c)
    41/54
  • d)
    54/41
Correct answer is option 'B'. Can you explain this answer?

Prerna Gupta answered
To find the value of (1/P + 1/Q), we need to determine the values of P and Q first.

Given that P and Q are the roots of the quadratic equation f(x) = x^2 - 14x + 45, we can use the quadratic formula to find their values.

The quadratic formula states that for a quadratic equation of the form ax^2 + bx + c = 0, the roots can be found using the formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = 1, b = -14, and c = 45. Plugging these values into the quadratic formula, we get:

P = (-(-14) ± √((-14)^2 - 4(1)(45))) / (2(1))
= (14 ± √(196 - 180)) / 2
= (14 ± √16) / 2
= (14 ± 4) / 2
= (18 / 2) or (10 / 2)
= 9 or 5

So, P can have the value of either 9 or 5.

Now, let's find the value of Q. Since P and Q are the roots of the quadratic equation, if P = 9, then Q = 5, and vice versa.

Now, we can calculate (1/P + 1/Q) using the values of P and Q that we found.

(1/P + 1/Q) = (1/9 + 1/5)

To add these fractions, we need a common denominator. The least common denominator of 9 and 5 is 45. We can rewrite the fractions with the common denominator:

(1/P + 1/Q) = (5/45 + 9/45)

Now, we can add the fractions:

(1/P + 1/Q) = (5 + 9) / 45
= 14 / 45

Therefore, the value of (1/P + 1/Q) is 14/45, which corresponds to option B.

Find the roots of the quadratic equation: x2 + 2x - 15 = 0?
  • a)
    -5, 3
  • b)
    3, 5
  • c)
    -3, 5
  • d)
    -3, -5
Correct answer is option 'A'. Can you explain this answer?

Vaishnavi Iyer answered
Solution:
The given quadratic equation is x^2 + 2x - 15 = 0.

To find the roots of the quadratic equation, we can use the quadratic formula which is given as:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

By comparing the given quadratic equation with ax^2 + bx + c = 0, we can see that a = 1, b = 2, and c = -15.

Substituting these values in the quadratic formula, we get:

x = (-2 ± √(2^2 - 4(1)(-15))) / 2(1)

x = (-2 ± √(64)) / 2

x = (-2 ± 8) / 2

Therefore, the roots of the quadratic equation are:

x = (-2 + 8) / 2 = 3

x = (-2 - 8) / 2 = -5

Hence, the correct answer is option A, which is -5 and 3.

The two numbers whose sum is 27 and their product is 182 are
  • a)
    12 and 13
  • b)
    12 and 15
  • c)
    14 and 15
  • d)
    13 and 14
Correct answer is option 'D'. Can you explain this answer?

Prateek Gupta answered
Explanation:Let the one number be xx .As the sum  of numbers is 27 , then the other number will be (27−x)(27−x)                                                                                                                                    According to question

For what value of b and c would the equation x2 + bx + c = 0 have roots equal to b and c.
  • a)
    (0,0)
  • b)
    (1,-2)
  • c)
    (1,2)
  • d)
    Both (a) and (b)
Correct answer is option 'D'. Can you explain this answer?

Aarav Sharma answered
To find the values of b and c that would make the equation x^2 + bx + c = 0 have roots equal to b and c, we can use the fact that the sum and product of the roots of a quadratic equation are related to its coefficients.

The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is given by -b/a, and the product of the roots is given by c/a.

In this case, we are given that the roots are b and c. So we have the following equations:

b + c = -b/a
bc = c/a

We can simplify the second equation by multiplying both sides by a:

abc = c

Now we have two equations:

b + c = -b/a
abc = c

We can solve these equations to find the values of b and c.

Solving the equations:
To solve the first equation, we can multiply both sides by a:

ab + ac = -b

Rearranging the equation, we get:

ab + b = -ac

Factoring out b, we have:

b(a + 1) = -ac

Dividing both sides by (a + 1), we get:

b = -ac/(a + 1)

Substituting this value of b into the second equation, we have:

abc = c

Replacing b with -ac/(a + 1), we get:

-a^2c/(a + 1) = c

Cross-multiplying, we have:

-a^2c = c(a + 1)

Simplifying, we get:

-a^2c = ac + c

Adding ac to both sides, we have:

ac - a^2c = c

Factoring out c, we get:

c(a - a^2) = c

Dividing both sides by c, we get:

a - a^2 = 1

This is a quadratic equation in terms of a. We can solve it by rearranging and factoring:

a^2 - a + 1 = 0

This equation does not have any real solutions. Therefore, there are no values of b and c that would make the equation x^2 + bx + c = 0 have roots equal to b and c.

Thus, the correct answer is option D) Both (a) and (b).

The sum of the squares of two consecutive positive integers exceeds their product by 91. Find the integers?
  • a)
    9, 10
  • b)
     10, 11
  • c)
    11, 12
  • d)
    12, 13
  • e)
    None of these
Correct answer is option 'A'. Can you explain this answer?

Dhruv Mehra answered
Let the two consecutive positive integers be x and x + 1
x2 + (x + 1)2 - x(x + 1) = 91
x2 + x - 90 = 0
(x + 10)(x - 9) = 0 => x = -10 or 9.
As x is positive x = 9
Hence the two consecutive positive integers are 9 and 10.

The product of two successive integral multiples of 5 is 1050. Then the numbers are
  • a)
    35 and 40
  • b)
    25 and 30
  • c)
    25 and 42
  • d)
    30 and 35
Correct answer is option 'D'. Can you explain this answer?

Prateek Gupta answered
Explanation:
Let one multiple of 5 be x then the next consecutive multiple of will be (x+5) According to question,
Then the number are 30 and 35.

I. a2 - 7a + 12 = 0,
II. b2 - 3b + 2 = 0 to solve both the equations to find the values of a and b?
  • a)
    if a < b
  • b)
    if a ≤ b
  • c)
    if the relationship between a and b cannot be established.
  • d)
    if a > b
  • e)
     if a ≥ b
Correct answer is option 'D'. Can you explain this answer?

Aarav Sharma answered
To solve both equations, we can use factoring or the quadratic formula.

I. a^2 - 7a + 12 = 0
We can factor this equation as:
(a - 3)(a - 4) = 0
Setting each factor equal to zero, we have:
a - 3 = 0 or a - 4 = 0
a = 3 or a = 4

So the values of a are a = 3 and a = 4.

II. b^2 - 3b + 2 = 0
This equation can also be factored as:
(b - 1)(b - 2) = 0
Setting each factor equal to zero, we have:
b - 1 = 0 or b - 2 = 0
b = 1 or b = 2

So the values of b are b = 1 and b = 2.

Therefore, the values of a are 3 and 4, and the values of b are 1 and 2.

I. a2 + 8a + 16 = 0,
II. b2 - 4b + 3 = 0 to solve both the equations to find the values of a and b?
  • a)
    If a < b
  • b)
    If a ≤ b
  • c)
    If the relationship between a and b cannot be established
  • d)
    If a > b
  • e)
    If a ≥ b
Correct answer is option 'A'. Can you explain this answer?

Sagar Sharma answered
I. To solve the equation a^2 + 8a + 16 = 0, we can use the quadratic formula:

a = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 8, and c = 16. Plugging these values into the quadratic formula:

a = (-8 ± √(8^2 - 4(1)(16))) / (2(1))

Simplifying:

a = (-8 ± √(64 - 64)) / 2

a = (-8 ± √0) / 2

a = -8 / 2

a = -4

So, the value of a is -4.

II. To solve the equation b^2 - 4b + 3 = 0, we can factorize it:

(b - 1)(b - 3) = 0

Setting each factor equal to zero:

b - 1 = 0 or b - 3 = 0

b = 1 or b = 3

So, the values of b are 1 and 3.

Therefore, the values of a and b are -4, 1, and 3.

For all x, x2 + 2ax + (10 − 3a) > 0, then the interval in which a lies, is?
  • a)
    a < -5
  • b)
    a > 5
  • c)
    -5 < a < 2
  • d)
    2 < a < 5
  • e)
    a < -2
Correct answer is option 'C'. Can you explain this answer?

Rahul Mehta answered
In f(x) = ax2 + bx + c
When a > 0 and D < 0
Then f(x) is always positive.
x2 + 2ax + 10 − 3a > 0, ∀x ∈ R
⇒ D < 0
⇒ 4a2 − 4(10 − 3a) < 0
⇒ a2 + 3a − 10 < 0
⇒ (a+5)(a−2) < 0
⇒ a ∈ (−5,2)

Chapter doubts & questions for Quadratic Equations - Mathematics for ACT 2025 is part of ACT exam preparation. The chapters have been prepared according to the ACT exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for ACT 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

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