Quadratic equations are everyday concepts with real-life applications. Understanding them is essential for solving aptitude and reasoning questions.

This article offers a variety of easy-to-understand quadratic equations questions. Whether you’re a student or want to enhance your aptitude and reasoning skills, these questions and explanations will help you improve your problem-solving abilities.

What is a Quadratic Equation?

To define in simple words , quadratic equation is a 2nd-degree equation with syntax as ax² + bx + c = 0 , where

• ‘x’ is an unknown variable
• a, b, and c’s are constants (real numbers)
• and a is not equal to 0.

To solve this kind of equation, you can use methods such as factoring, completing the square, or quadratic formula (also known as the Shree Dharacharya formula) . Each method helps find the values of x that satisfy the equation.

Related Articles:

• Quadratic Equations
• Quadratic Formula
• Roots of Quadratic Equation

Important Formulas / Concepts

• Standard Form of Quadratic Equation : ax² + bx + c = 0 where ‘x’ is an unknown variable , ‘a, b, and c’ are constants (real numbers) and a is not equal to 0.
• Roots of Quadratic Equation Formula : x = [- b ± √(b² – 4ac)]/2a and b² – 4ac = D is called the Determinant of the quadratic equation.
• Range of quadratic expression : Maximum and Minimum value of y = ax² + bx + c = 0 occurs at x = – (b/2a) irrespective of a < 0 or a > 0 respectively .
• Sum of Roots of Quadratic Equation : Sum of the roots α + β is equal to -b/a ( = – Coefficient of x / Coefficient of x²)
• Product of Roots of Quadratic Equation : Product of roots α . β is equal to c/a​ ( = Constant term / Coefficient of x² )
• Difference of Roots of Quadratic Equation : Difference of roots α – β is equal to √D/a .
• Relation between roots and coefficient : (x – α) (x – β ) = 0 or x² -(sum of roots) x + product of roots = 0.

Practice Questions on Quadratic Equations : Solved

1. Solve the quadratic equation using factorization : x² – 4x + 4 = 0.

x² – 4x + 4 = 0
[Tex]\Rightarrow[/Tex] x²– 2x – 2x + 4 = 0
[Tex]\Rightarrow[/Tex] x(x – 2) – 2 (x – 2) = 0
[Tex]\Rightarrow[/Tex] (x – 2)(x – 2) = 0
[Tex]\Rightarrow[/Tex] (x – 2)² = 0
Therefore x = 2

2. Form a quadratic equation with rational coefficients if one of its root is cot²18°

Given one of its root is cot²18°
Then , cot²18° = (1 + cos 36° )/(1 – cos 36°)
[Tex]\Rightarrow[/Tex] (1 + (√5 + 1)/4)/(1 – (√5 + 1)/4)
[Tex]\Rightarrow[/Tex] 5 + 2√5
Hence if α = 5 + 2√5 , β = 5 – 2√5
Therefore , α + β = 10 ; α.β = 25 – 20 = 5
So , the required quadratic equation will be x² – 10x + 5 = 0

3. One root of mx² – 10x + 3 = 0 is two third of the other root . Find the sum of the roots.

α + 2α/3 = 10/m
[Tex]\Rightarrow[/Tex] 5α/3 = 10/m
[Tex]\Rightarrow[/Tex] α = 6/m

and 2α/3 = 3/m
[Tex]\Rightarrow[/Tex] 2α² = 9/m
[Tex]\Rightarrow[/Tex] 2.36/m² = 9/m
[Tex]\Rightarrow[/Tex] m = 8

Therefore , Sum = 10/ m = 10/8 = 5/4.

4. Form a quadratic equation with roots 2 and 3.

Sum of roots = 2 + 3 = 5
Product of roots = 2.3 = 6

Therefore , quadratic equation is given by x² + (sum of roots)x + (product of roots) = 0
So , the required equation is x² + 5x + 6 = 0.

5. If x = 1 and x = 2 are solutions of the equation x³ + ax²+ bx + c = 0 and a + b = 1, then find the value b.

a + b + c = -1 so, c = -2
and 8 + 4a + 2b + c = 0
[Tex]\Rightarrow[/Tex] 4a + 2b = -6 [Tex]\Rightarrow[/Tex] 2a + b = -3
[Tex]\Rightarrow[/Tex] a = -4 , b = 5

Hence , a = -4, b = 5 and c = -2.

6.Find the roots of the equation x⁴ + x³ – 19x² – 49x – 30 = 0 , given that the roots are all rational numbers.

Since all the roots are rational because , they are the divisors of -30.
The divisors of -30 are 1, 2, 3, 4, 5, 6, 10, 15, 30 and -1,-2,-3,-4,-5,-6,-10,-15,-30.
By remainder theorem , we find that -1,-2,-3 and 5 are the roots .
Hence the roots are -1,-2,-3 and +5.

Practice Questions on Quadratic Equations : Unsolved

1. Solve: 9 + 7x = 7x²
2. If one root is twice of the other , find the quadratic equation .
3. Difference of roots is 2 and their sum is 7 , find the quadratic equation .
4. One root of mx² – 10x + 3 = 0 is two third of the other root . Find the product of the roots.
5. If the product of the roots of the equation mx² + 6x + 2m – 1 = 0 is -1 then find m .
6. For what value of a , the difference of the roots of the equation (a – 2)x² – (a – 4)x – 2 = 0 is equal to 3.
7. For what value of a the sum of the roots of the equation x² + 2(2 – a – a²)x – a² = 0 is zero .
8. The number of roots of the quadratic equation 8sec²x – 6secx + 1 = 0.
9. If the roots of the equation 6x² – 7x + k = 0 are rational , then find k.
10. If x is real then find the maximum value of (3x² + 9x + 17)/(3x² + 9x + 7).

FAQs – Practice Questions on Quadratic Equations

What is meant by ‘quadratic equation’?

A polynomial equation of the 2nd-degree in one variable is called the quadratic equation. The general form of the quadratic equation looks like ax² + bx + c = 0 , where a and b are the coefficients of x² and x terms respectively, and c is the constant term and a , b , c are real . Note that a can’t be equal to zero.

What is standard form of quadratic equation?

Standard Form of Quadratic Equation is ax² + bx + c = 0

What is quadratic formula?

General formula to solve a quadratic equation of the form ax² + bx + c = 0 is

[Tex]x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} [/Tex]

Give some methods to solve quadratic equations.

To solve this kind of equation, you can use methods such as factoring, completing the square, or quadratic formula (also known as the Shree Dharacharya formula) . Each method helps find the values of x that satisfy the equation.

How to factor a quadratic equation?

To factor a quadratic equation of the form, ax² + bx +c =0:

• Step 1 :Find two numbers that multiply to ac (the product of the coefficient of x² and the constant term) and add to b (the coefficient of x).
• Step 2 :Rewrite the middle term (bx) as the sum of two terms using the numbers found in step 1.
• Step 3 :Factor by grouping.

How does quadratic equation differs from linear equation?

Quadratic Equation differs from Linear Equation in the manner that Quadratic Equation has degree two and can have maximum of two solutions while Linear Equation has degree one and can have only one solution at maximum.

What are the roots of quadratic equation?

The roots of a quadratic equation are the values of x that satisfy the equation ax² + bx + c = 0 where, a, b, and c are Real Numbers and Constants and a ≠ 0.