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.

• 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