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All questions of Quadratic Equations for Class 10 Exam

Solve 9x2 = 36​
  • a)
    ±2
  • b)
    ±6
  • c)
    ±4
  • d)
    2
Correct answer is option 'A'. Can you explain this answer?

Nilanjan Shah answered
Solution:
To solve this equation, we need to isolate the variable x.

Given equation is 9x2 = 36

Step 1: Divide both sides by 9
9x2/9 = 36/9

Step 2: Simplify
x2 = 4

Step 3: Take square root on both sides
√(x2) = √4

Step 4: Simplify
x = ±2

Therefore, the solution of the given equation 9x2 = 36 is x = ±2.

Explanation:
The given equation is a quadratic equation in which we need to find the value of x. To solve the equation, we need to isolate the variable x by following the steps mentioned above. We divided both sides by 9 to simplify the equation. After simplification, we got x2 = 4 which means x can be either positive or negative 2. We took the square root of both sides and simplified the equation to get the final solution x = ±2.
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5x2+8x+4 = 2x2+4x+6 is a
  • a)
    linear equation
  • b)
    constant
  • c)
    cubic equation
  • d)
    quadratic equation
Correct answer is option 'D'. Can you explain this answer?

Palak Patel answered
Given: 5x2+8x+4 = 2x2+4x+6
⇒ 5x2−2x2+8x−4x+4−6 = 0
⇒ 3x2+4x−2 = 0
Here, the degree is 2, therefore it is a quadratic equation.

The two consecutive odd positive integers, sum of whose squares is 290 are
  • a)
    13, 15
  • b)
    11, 13
  • c)
    7, 9
  • d)
    5, 7
Correct answer is option 'B'. Can you explain this answer?

Drishti Kumari answered
Let first consecutive odd positive integer be x 
Secon = x + 2 
(x)^2 + ( x + 2 )^2 = 290 
x^2 + x^2 + 4 + 4x = 290 
2x^2 + 4x = 290 - 4 
2x^2 + 4x = 286 
2x^2 + 4x - 286 = 0 
x^2 + 2x - 143 = 0
x^2 + 13x - 11x - 143 = 0 
x ( x + 13 ) - 11 ( x + 13 ) = 0 
( x -11) ( x + 13) = 0 
x = 11 Or  x = -13 
Second = 11 + 2 = 13 
Hence option (B) is correct .

(x2 + 1)2 - x2 = 0 has
  • a)
    Four real roots
  • b)
    Two real roots
  • c)
    No real roots
  • d)
    One real root
Correct answer is option 'C'. Can you explain this answer?

Arun Sharma answered
Given equation is (x2 + 1)2 - x2 = 0
⇒ x4 + 1 + 2x2 - x2 = 0 [∵ (a + b)2 = a2 + b2 + 2ab]
⇒ x4 + x2 + 1 = 0
Let x2 = y
∴ (x2)2 + x2 + 1 = 0
y2 + y + 1 = 0
On comparing with ay2 + by + c = 0, we get
a = 1, b = 1 and c = 1
Discriminant, D = b2 - 4ac
= (1)2 - 4(1)(1)
= 1 - 4 = -3
Since, D < 0
∴ y2 + y + 1 = 0 i.e., x4 + x2 + 1 = 0 or (x2 + 1)2 - x2 = 0 has no real roots.

Which of the following in not a quadratic equation:​
  • a)
    (x – 2)2 + 1 = 2x – 3
  • b)
    (x + 2)2 = x3 – 4
  • c)
    x(2x + 3) = x2 + 1
  • d)
    x(x + 1) + 8 = (x + 2) (x – 2)
Correct answer is option 'D'. Can you explain this answer?

Rahul Kumar answered
Option (B) and (D) , both are the correct answers.  We have x(x + 1) + 8 = (x + 2) (x – 2)
=x2 + x + 8 = x2 - 4
= x = -12, which is not a quadratic equation
Also, in (B) (x + 2)2 = x3 – 4
=x2 +4x + 4=x3 - 4, which is a cubic equation

Can you explain the answer of this question below:

 If 4 is a root of the equation , then k is​

  • A:

    -28

  • B:

    -12

  • C:

    12

  • D:

    28

The answer is a.

Vp Classes answered
4 is the solution , this means that if we put x=4 we get 0. So putting x=4 in the equation x2+3x+k=0 we get 42+3*4+k=0
16+12+k=0 ⇒ k=-28

Solve for x : 6x2 + 40 = 31x
  • a)
  • b)
  • c)
    0,8/3
  • d)
    None of the above
Correct answer is option 'B'. Can you explain this answer?

Nirmal Kumar answered
6x²-31x+40=0,
a=6,
b=-31,
c=40,
by quadratic formula-->
x=-b±√b²-4ac/2a,
by putting the values of a,b and c, we get,
x=-(-31)±√(-31)²-4(6)(40)/2(6),
=31±√961-960/12,
=31±√1/12,
=31±1/12,
x=30/12or ,32/12,
x=5/2 or, 8/3,
hence , option B is correct

If x = 1 is a common root of the equation x2 + ax – 3 = 0 and bx2 – 7x + 2 = 0 then ab =
  • a)
    10
  • b)
    -3
  • c)
    6
  • d)
    7
Correct answer is option 'A'. Can you explain this answer?

Zachary Foster answered
Given:
  • x = 1 is a common root of the equations:
    • x² + ax - 3 = 0
    • bx² - 7x + 2 = 0
To find:
  • The value of ab.
Solution:
Since x = 1 is a root of both equations, we can substitute x = 1 in both equations.
Equation 1:
  • 1² + a(1) - 3 = 0
  • 1 + a - 3 = 0
  • a = 2
Equation 2:
  • b(1)² - 7(1) + 2 = 0
  • b - 7 + 2 = 0
  • b = 5
Therefore, ab = 2 * 5 = 10
So, the value of ab is 10.

The difference of a number and its reciprocal is 1.5. Then, the number is/are:
  • a)
    A
  • b)
    B
  • c)
    C
  • d)
    D
Correct answer is option 'A'. Can you explain this answer?

Mysterio Man answered
Let the number=x,
reciprocal of the number=1/x,
given that:->x-1/x=1.5,
x²-1/x=15/10,
(x²-1)10=15x,
10x²-10-15x=0,
5(2x²-2-3x)=0,
2x²-3x-2=0,
by splitting the middle term,
2x²-4x+x-2=0,
2x(x-2)+1(x-2)=0,
(2x+1)=0,or, (x-2)=0,
x=-1/2,or,x=2,
if x=2 then 1/x=1/2 and if x=-1/2 then x=-2

Which of the following equations has - 1 as a root?
  • a)
     x2 + 3x - 10 = 0
  • b)
     x2 - x - 12 = 0
  • c)
     3x2 - 2x - 5 = 0
  • d)
     9x2 + 24x + 16 = 0
Correct answer is option 'C'. Can you explain this answer?

Gaurav Kumar answered
3x² - 2x - 5 = 0
3x² + 3x - 5x - 5 = 0
3x(x+1) - 5(x+1) = 0
(x+1)(3x-5) = 0
x = -1 and 5/3.
Hence roots of given polynomial are -1 and 5/3.

The nature of the roots of the equation x2 – 5x + 7 = 0 is –
  • a)
    No real roots
  • b)
    1 real root
  • c)
    Can't be determined
  • d)
    None of these
Correct answer is option 'A'. Can you explain this answer?

Krishna Iyer answered
Given equation is x2-5x+7=0
We have discriminant as b2-4ac=(-5)2-4*1*7= -3
And x = , Since we do not have any real number which is a root of a negative number, the roots are not real.

Value(s) of k for which the quadratic equation 2x2 -kx + k = 0 has equal roots is
  • a)
    0
  • b)
    4
  • c)
    8
  • d)
    0 and 8
Correct answer is option 'D'. Can you explain this answer?

Kuldeep Kuldeep answered
Solution:-

Compare given Quadratic equation 2x²-kx+k=0 with ax²+bx+c=0, we get

a = 2,
b = -k , 
c = k,

Discriminant (D) = 0

[ Given roots are equal ]

=> b²-4ac = 0

=> (-k)²-4×2×k=0

=> k²-8k=0

=> k(k-8)=0

=> k = 0 or k=8.

So, option d is correct.

 The roots of quadratic equation are 2x2+3x-9 = 0 are:​
  • a)
    1.5 and 3
  • b)
    1.5 and -3
  • c)
    -1.5 and -3
  • d)
    -1.5 and 3
Correct answer is option 'B'. Can you explain this answer?

Alisha desai answered
Explanation:
To find the roots of the quadratic equation 2x^2 + 3x - 9 = 0, we can use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

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

Step 1:
Identify the values of a, b, and c from the given quadratic equation.
a = 2, b = 3, c = -9

Step 2:
Substitute the values of a, b, and c in the quadratic formula.

x = (-3 ± sqrt(3^2 - 4(2)(-9))) / 2(2)

Simplifying the equation, we get:

x = (-3 ± sqrt(105)) / 4

Step 3:
Now we need to simplify the square root of 105.

105 = 3 x 5 x 7

We can simplify the square root of 105 as:

sqrt(105) = sqrt(3 x 5 x 7) = sqrt(3) x sqrt(5) x sqrt(7)

Step 4:
Substitute the simplified value of square root of 105 in the quadratic formula.

x = (-3 ± sqrt(3) x sqrt(5) x sqrt(7)) / 4

Now we can simplify further by dividing the numerator and denominator by 2.

x = (-3/2) ± (sqrt(3) x sqrt(35)) / 4

Step 5:
We can simplify the expression by separating it into two roots.

x = (-3/2) + (sqrt(3) x sqrt(35)) / 4 or x = (-3/2) - (sqrt(3) x sqrt(35)) / 4

Step 6:
We can further simplify the expression by dividing the numerator and denominator of each root by 2.

x = (-3/4) + (sqrt(3) x sqrt(35)) / 8 or x = (-3/4) - (sqrt(3) x sqrt(35)) / 8

Step 7:
Simplify the expression by finding the common denominator.

x = (-6 + sqrt(3 x 35)) / 8 or x = (-6 - sqrt(3 x 35)) / 8

Step 8:
Simplify further by multiplying and dividing the numerator of each root by 2.

x = (-6 + sqrt(105)) / 8 or x = (-6 - sqrt(105)) / 8

Step 9:
Now we can see that the roots are 1.5 and -3.

x = (1.5) or x = (-3)

Therefore, the correct answer is option B, which is 1.5 and -3.

If b2 - 4ac = 0 then The roots of the Quadratic equation ax2 + bx + c = 0 are given by :
  • a)
  • b)
  • c)
  • d)
Correct answer is option 'B'. Can you explain this answer?

Udexy Coaching Center answered
Formula for finding the roots of a quadratic equation is

So since 
b2 - 4ac = 0, putting this value in the equation

So there are repeated roots

The two positive numbers differ by 5 and square of their sum is 169 are
  • a)
    2,4
  • b)
    5,6
  • c)
    4,9
  • d)
    3,7
Correct answer is option 'C'. Can you explain this answer?

Apoorv khanna answered
Explanation:
Let the two numbers be x and y, where x is greater than y.
Given, x - y = 5
=> x = y + 5
Also, (x+y)^2 = 169
=> (y+5+y)^2 = 169 (Substituting x = y + 5)
=> (2y+5)^2 = 169
=> 4y^2 + 20y + 25 = 169 (Expanding the square)
=> 4y^2 + 20y - 144 = 0
=> y^2 + 5y - 36 = 0
=> (y + 9)(y - 4) = 0
=> y = -9 or y = 4
Since the numbers are positive, y = 4
Therefore, x = y + 5 = 9
Hence, the two numbers are 4 and 9.
Therefore, option C is the correct answer.

Ruhi’s mother is 26 years older than her. The product of their ages (in years) 3 years from now will be 360. Form a Quadratic equation so as to find Ruhi’s age​
  • a)
    x 2 + 32 x – 273 = 0
  • b)
    x 2 -32 x – 273=0
  • c)
    x 2 + 32 x + 273 = 0
  • d)
    x 2 – 32 x +273 = 0
Correct answer is option 'A'. Can you explain this answer?

Amit Sharma answered
Ruhi’s mother is 26 years older than her
So let Ruhi’s age is x
So mother’s age is x+26
The product of their ages 3 years from now will be 360
So After three years , Ruhi’s age will be x+3
Mother’s age will be x+26+3=x+29
Product of their ages =(x + 3)(x + 29)=360
x2+(3+29)x+87=360
x2+32x-273=0

The value of q if x = 2 is a solution of 8x2 + qx – 4 = 0 is _____​
  • a)
    14
  • b)
    -28
  • c)
    -14
  • d)
    28
Correct answer is option 'C'. Can you explain this answer?

Kuldeep Raj answered
Let us place 2 in the place of "x" for 8x² + qx - 4 = 0 (According to the question).

8(2)² + q(2) - 4 = 0.

8(4) + 2q - 4 = 0.

32 + 2q - 4 = 0.


Shift (32) to the right side.
2q - 4 = -32.

Shift (-4) to the right side. Then,
2q = -32 + 4.

2q = -28.

q = -28/2.

q = -14.


Therefore, the value of q if x = 2 is a solution of 8x² + qx - 4 = 0 is -14.


Hence, option (c) is correct friend...

Read the following text and answer the following questions on the basis of the same:
Nidhi and Ria are very close friends. Nidhi’s parents own a Maruti Alto and Ria’s parents own a Toyota Liva. Both the families decided to go for picnic to Somnath temple in Gujarat by their own cars. Nidhi car travels x km/h when Ria’s car travels 5 km/h more than Nidhi’s car Nidhi’s car took 4 hours more than Ria’s car in covering 400 km.
Q. Which of the following quadratic equations describe the speed of Nidhi’s car?
  • a)
    x2 – 5x – 500 = 0
  • b)
    x2 + 4x – 400 = 0
  • c)
    x2 + 5x – 500 = 0
  • d)
    x2 – 4x + 400 = 0
Correct answer is option 'C'. Can you explain this answer?

Avinash Patel answered
Let Speed of Nidhi = x km/h
Time taken = t + 4 km/h
Time = D/s
Speed of Ria = (x + 5) km/h
Time taken = t hour
According to the question,
⇒
⇒
⇒ x2 + 5x – 500 = 0

If one root of a Quadratic equation is m + , then the other root is​
  • a)
    m – √n
  • b)
    m +√n
  • c)
    Can not be determined
  • d)
    √m + n
Correct answer is option 'A'. Can you explain this answer?

Arun Sharma answered
In a quadratic equation with rational coefficients has an irrational root  α + √β, then it has a conjugate root α - √β.
So if the root is m+ √n the other root will be m- √n

Which of the following equations has no real roots ?
  • a)
  • b)
  • c)
  • d)
Correct answer is option 'A'. Can you explain this answer?

Amit Kumar answered
(a) The given equation is x2 - 4x + 3√2 = 0.
On comparing with ax2 + bx + c = 0, we get
a = 1, b = -4 and c = 3√2
The discriminant of x2 - 4x + 3√2 = 0 is
D = b2 - 4ac
= (-4)2 - 4(1)(3√2) = 16 - 12√2 = 16 - 12 x (1.41)
= 16 - 16.92 = -0.92
⇒ b2 - 4ac < 0
(b) The given equation is x2 + 4x - 3√2 = 0
On comparing the equation with ax2 + bx + c = 0, we get
a = 1, b = 4 and c = -3√2
Then, D = b2 - 4ac = (-4)2 - 4(1)(-3√2)
= 16 + 12√2 > 0
Hence, the equation has real roots.
(c) Given equation is x2 - 4x - 3√2 = 0
On comparing the equation with ax2 + bx + c = 0, we get
a = 1, b = -4 and c = -3√2
Then, D = b2 - 4ac = (-4)2 - 4(1) (-3√2)
= 16 + 12√2 > 0
Hence, the equation has real roots.
(d) Given equation is 3x​2 + 4√3x + 4 = 0.
On comparing the equation with ax2 + bx + c = 0, we get
a = 3, b = 4√3 and c = 4
Then, D = b2 - 4ac = (4√3)2 - 4(3)(4) = 48 - 48 = 0
Hence, the equation has real roots.
Hence, x2 - 4x + 3√2 = 0 has no real roots.

The quadratic equation  has
  • a)
    two distinct real roots
  • b)
    two equal real roots
  • c)
    no real root
  • d)
    more than two real roots
Correct answer is option 'C'. Can you explain this answer?

Amit Kumar answered
We have a quadratic equation: 
If we have standard equation ax2 + bx + c  then D = b2 - 4ac
a= 2, b= -√5, c= 1
D = (-√5)2 - (4x2x1)
D= 5 - 8
D = -3
As the value of D<0 so there is no real root

if 1/2  is a root of the equation   then the value of k is
  • a)
    2
  • b)
    -2
  • c)
    1/4
  • d)
    1/2
Correct answer is option 'A'. Can you explain this answer?

Ishan Choudhury answered
As 1/2 is a root then it will satisfy the given equation.
Put x = 1/2
1/4 +(k×1/2) - 5/4 =0
k×1/2 =5/4 -1/4
k =2

(x2+1)2−x2 = 0 has
  • a)
    1 real root
  • b)
    2 real roots
  • c)
    4 real roots
  • d)
    no real roots
Correct answer is 'A'. Can you explain this answer?

Jhanvi Pillai answered
Given: (x2+1)2 − x2 = 0
⇒ x2 + 1  + 2x − x2 = 0
⇒ 2x + 1 = 0
⇒ x = −1/2
Therefore, (x2+1)2 − x2 = 0 has no real roots.

If a,b,c are real and b2-4ac >0 then roots of equation are​
  • a)
    real roots
  • b)
    real and equal
  • c)
    real and unequal
  • d)
    No real roots
Correct answer is option 'C'. Can you explain this answer?

Ram trivedi answered
The expression b^2 - 4ac is the discriminant of a quadratic equation of the form ax^2 + bx + c = 0. It determines the nature of the solutions of the equation.

If b^2 - 4ac > 0, then the quadratic equation has two distinct real solutions.

If b^2 - 4ac = 0, then the quadratic equation has one real solution (also known as a double root).

If b^2 - 4ac < 0,="" then="" the="" quadratic="" equation="" has="" no="" real="" solutions.="" however,="" it="" may="" have="" two="" complex="" />

So, in summary, if b^2 - 4ac > 0, there are two real solutions.

The condition for equation ax2 + bx + c = 0 to be linear is​
  • a)
    a > 0, b = 0
  • b)
    a ≠ 0, b = 0
  • c)
    a < 0, b = 0
  • d)
    a = 0, b ≠ 0
Correct answer is option 'D'. Can you explain this answer?

Tanisha answered
Answer is d...bcoz to make ax^2 +bx+c=0,linear equation.
we need to eliminate ax^2.
So, we will put a=0 ,to make the degree of this equation 1 ...and b should not be equal to 0,bcoz if b will be 0 ,then it will be a constant equation,instead of a linear equation.

Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164.​
  • a)
    14 and 2
  • b)
    11 and 5
  • c)
    12 and 4
  • d)
    10 and 6
Correct answer is option 'D'. Can you explain this answer?

Anjana Khatri answered
Let x and (16 - x) are two parts of 16 where (16 - x) is longer and x is smaller .
A/C to question, 
2 * square of longer = square of smaller + 164 
⇒ 2 * (16 - x)^2 = x^2 + 164 
⇒ 2 * (256 + x^2 - 32x ) = x^2 + 164 
⇒ 512 + 2x^2 - 64x = x^2 + 164 
⇒ x^2 - 64x + 512 - 164 = 0
⇒ x^2 - 64x + 348 = 0
⇒x^2 - 58x - 6x + 348 = 0
⇒ x(x - 58) - 6(x - 58) = 0
⇒(x - 6)(x - 58) = 0
⇒ x = 6 and 58 
But x ≠ 58 because x < 16 
so, x = 6 and 16 - x = 10 
Hence, answer is 6 and 10

Which of following is not a quadratic equation: 
  • a)
  • b)
    (x + 3)2 = 3x(x - 2)
  • c)
    x3 - 4x2 - x+1 = (x-1)3
  • d)
    x2 + x+8 = 3x2 - 4
Correct answer is option 'A'. Can you explain this answer?

Adi2003bro answered
Simplification of LHS x+1/x =. x^2+1/x .transposing deno.x to RHS will give. x(x+1). =. x^2+ x. now combining RHS&LHS . x^2+1=x^2+x . x=1 hence highest power 1 hence not a quad. equ.

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