Practice Questions on Quadratic Equations

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

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 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 the quadratic formula (also known as the Shree Dharacharya formula).

Solved Questions

Q1. Solve the quadratic equation using factorization: x² - 4x + 4 = 0.

x² - 4x + 4 = 0
x²- 2x - 2x + 4 = 0
x(x - 2) - 2 (x - 2) = 0
(x - 2)(x - 2) = 0
(x - 2)² = 0
Therefore x = 2

Q2. 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°)
(1 + (√5 + 1)/4)/(1 - (√5 + 1)/4)
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

Q3. 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
5α/3 = 10/m
α = 6/m

and 2α/3 = 3/m
2α² = 9/m
2.36/m² = 9/m
m = 8

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

Q4. 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.

Q5. 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
4a + 2b = -6 2a + b = -3
a = -4 , b = 5

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

Q6. 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

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).

Related Articles

• Quadratic Equations
• Quadratic Formula
• Roots of Quadratic Equation