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# Quadratic Equations Questions and Answers

## Questions on Quadratic Equations

If you’re looking to master quadratic equations, understanding and practicing various questions on this topic is essential. Quadratic equations are second-degree polynomial equations with the general form ax^2 + bx + c = 0, where ‘a’, ‘b’, and ‘c’ are constants. These equations have two solutions, called roots, which can be found using methods like factoring, completing the square, or using the quadratic formula.

### Methods for finding roots of Quadratic Equation

The roots or solutions of a quadratic equation are the values of ‘x’ that satisfy the equation, making it true. A quadratic equation can have two real roots, two complex roots, or one real root (in case of a perfect square). The number of roots is determined by the value of the discriminant (Δ) given by:

Δ = $b^{2}-4ac$

1. If Δ > 0, the quadratic equation has two distinct real roots.
2. If Δ = 0, the quadratic equation has one real root (repeated or equal roots).
3. If Δ < 0, the quadratic equation has two complex roots (conjugate pairs).

To find the roots of a quadratic equation, several methods can be used:

1. Factoring: If the equation can be factored, the roots can be directly determined from the factors.

2. Quadratic Formula: The quadratic formula is a general method to find the roots of any quadratic equation and is given by:

$x = \frac{(-b\pm \sqrt{b^{2}-4ac})}{2a}$

where the ± sign accounts for the two possible solutions.

3. Completing the Square: This method involves transforming the quadratic equation into a perfect square form, from which the roots can be easily obtained.

### Rules for Quadratic Equations

• If p+ √q is a root of a quadratic equation, then its other root is p-√q.
• When D ≥ 0, then rx^2+ dx + z can be expressed as a product of two linear factors.
• If α and β are the roots of rx^2+ dx + z, then we can write it as: x^2 – (α + β)x + α β = 0.

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Question 1

Time: 00:00:00
Calculate the number of real roots of the quadratic equation $a^{2}+5a+6= 0$ ?

7

7

1

1

4

4

2

2

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Question 2

Time: 00:00:00
Calculate the number of real roots of the equation $a^{4}+ a^{2}+1= 0$

2

2

-3

-3

-2

-2

None

None

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Question 3

Time: 00:00:00
Calculate the number of distinct real roots of the equation $p^{11}+7p^{3}-5=0$

-3

-3

1

1

-6

-6

2

2

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Question 4

Time: 00:00:00
Calculate the value of p when the equation $a^{2}-(p+6) a+2(2p-1)= 0$ if the total of roots equals half their product?

7

7

-7/3

-7/3

1

1

zero

zero

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Question 5

Time: 00:00:00
Which of the following is a quadratic equation?

x³ + 2x + 1 = x² + 3

x³ + 2x + 1 = x² + 3

– 2x = (5 – x) (2x – 25)

– 2x = (5 – x) (2x – 25)

( k + 1 )x² +32x=7, where k = -1

( k + 1 )x² +32x=7, where k = -1

x³x² = (x³)

x³x² = (x³)

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Question 6

Time: 00:00:00
Which of the following is the solution to the inequality 2x^2 + 5x - 3 > 0?

x > 3/2 or x < -1

x > 3/2 or x < -1

x > 3/2 and x < -1

x > 3/2 and x < -1

x > -3/2 and x < 1

x > -3/2 and x < 1

x < -3 and x > 1/2

x < -3 and x > 1/2

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Question 7

Time: 00:00:00
Calculate the sum of roots of the below mentioned quadratic equation:

$3a^{2}$+2a-1= 0.

−2/3

−2/3

1/4

1/4

7/3

7/3

-5/6

-5/6

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Question 8

Time: 00:00:00
Calculate the product of roots of the below mentioned quadratic equation:

$3a^{2}$+2a-1= 0.

2/3

2/3

zero

zero

-1/3

-1/3

1/7

1/7

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Question 9

Time: 00:00:00
What will be the roots of the equation p2m2+pqm= q2 where p≠0

Real

Real

Not real

Not real

Real but unequal

Real but unequal

None of the above

None of the above

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Question 10

Time: 00:00:00
The quadratic equations x²-5x+3=0 has

no real roots

no real roots

more than two real roots

more than two real roots

two equal real roots

two equal real roots

two distinct real roots

two distinct real roots

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Question 11

Time: 00:00:00
For what value of p , equation 2a+3b= 5 and 4a+pb= Q will not have any solution?

1

1

Zero

Zero

2

2

6

6

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Question 12

Time: 00:00:00
IF p and q are the roots of the equation b2-13b+f= 0 and p-q= 3 then calculate the value of f.

40

40

71

71

74

74

53

53

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Question 13

Time: 00:00:00
Question 3: The volume of a rectangular box (rectangular prism) is given by $V = 36x^{3} - 30x^{2} + 5x$, where 'x' is the length of each side. What is the value of 'x' that maximizes the volume?

x = 2/3

x = 2/3

x = 1/6

x = 1/6

x = 3/4

x = 3/4

x = 1

x = 1

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Question 14

Time: 00:00:00
Calculate the value of p, if f(p)= P2 – 28p+196 and f(p)= p

74

74

14

14

52

52

12

12

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Question 15

Time: 00:00:00
Calculate the value of ‘a’ if p+3 is a  factor of 3p2+ap+6?

11

11

44

44

-21

-21

None of the above

None of the above

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Question 16

Time: 00:00:00
If a+2 is the factor of equation a4-4a2+2pa+3= 0 then calculate the value of p?

3/4

3/4

6

6

9

9

1/2

1/2

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Question 17

Time: 00:00:00
p2-5p+6= 0 and p2- a p+3= 0 have a common root. Then calculate the value of a?

4

4

-4

-4

5

5

None of the above

None of the above

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Question 18

Time: 00:00:00
Find the number of integral roots of equation a3-a2+2a-17= 0

8

8

4

4

3

3

None of the above

None of the above

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Question 19

Time: 00:00:00
Calculate the number of the real solution of a- (1/a² - 4) = 2- (1/a² - 4)

7

7

6

6

1

1

Zero

Zero

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Question 20

Time: 00:00:00
Calculate the total of real roots of quadratic equation a2+a-6= 0

Zero

Zero

9

9

2

2

1

1

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Question 21

Time: 00:00:00
How many numbers in the set (-4, -3, 0, 2) fulfil the condition (x-4)<6 and (x+4)<5

-1

-1

1

1

-7

-7

None of the above

None of the above

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Question 22

Time: 00:00:00
If a and b are the roots of equation p2+p+2= 0 then $\frac{a^{10} + b^{10}}{a^{-10} + b^{-10}}$ would be equivalent to which number?

1024

1024

4123

4123

741

741

1000

1000

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Question 23

Time: 00:00:00
Calculate the birth year of a woman who was born in the 1900s, and in 1980, she figured out that her age was the square root of her birth year?

1936

1936

1990

1990

1970

1970

1950

1950

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Question 24

Time: 00:00:00
Solve the inequation m3- m2 +m -1 >0

m>1

m>1

m<1

m<1

m>7

m>7

m>-2

m>-2

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Question 25

Time: 00:00:00
Calculate the total number of real roots of the equation p2+5p+6= 0

5

5

-1

-1

2

2

None of the above

None of the above

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Question 26

Time: 00:00:00
If p= 2+21/3 then what will be the value of p3-6p2+6p

Zero

Zero

-5

-5

1

1

2

2

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Question 27

Time: 00:00:00
If a1, a2, and a3 are in the AP and the coefficient of a in the equation with a1, and a2,as its root is -4, then find one of the root?

7

7

4

4

3

3

Cannot be determined

Cannot be determined

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Question 28

Time: 00:00:00
If p1, p2, and p3 are in the AP and the coefficient of ‘p’ in the equation with p1, and p2, as its root is -6, then find one of the root?

9

9

5

5

Zero

Zero

Cannot be determined

Cannot be determined

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Question 29

Time: 00:00:00
For p, q, and r € X if pa2+qa+r is real for the real values of a and imaginary for the imaginary value of a then what will be the value of p?

P= 1

P= 1

P>9

P>9

P= 0

P= 0

None of the above

None of the above

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Question 30

Time: 00:00:00
Calculate the number that exceeds its reciprocal by 35/6

8

8

6

6

41

41

2

2

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["0","40","60","80","100"]
["Need more practice!","Keep trying!","Not bad!","Good work!","Perfect!"]

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