Worksheet on Quadratic Equations is provided here. Students can practice multiple questions on finding the roots of a quadratic equation and check whether the given equation is quadratic or not. The best ways to solve the quadratic equation is by factorization, using the formula, and by drawing a graph. Practice more questions on the Quadratic Equations Worksheet for free and gain knowledge on this concept.
We are suggesting you to refer the Quadratic Formula Worksheet with Answers and begin preparation. Mostly use the factorization method to get the roots of the equation easily.
1. Which of the following are quadratic equations?
(i) 5x² + √8x + 5 = 0
(ii) x + 6/x = x
(iii) x² = 0
(iv) x² – 4x = 0
(v) x² – 25 = 0
Solution:
(i) 5x² + √8x + 5 = 0 is in the form of ax² + bx + c = 0. So, it is a quadratic equation.
(ii) x + 6/x = x
(x² + 6)/x = x
x² + 6 = x²
Which is not in the form of ax² + bx + c = 0. So, it is not a quadratic equation.
(iii) x² = 0 is in the form of ax² + bx + c = 0. So, it is a quadratic equation.
(iv) x² – 4x = 0 is in the form of ax² + bx + c = 0. So, it is a quadratic equation.
(v) x² – 25 = 0 is in the form of ax² + bx + c = 0. So, it is a quadratic equation.
2. Find which of the following are quadratic equations?
(i) x² = 64
(ii) x² + 4x + 2 = 0
(iii) x² + 3/x² = 8
(iv) x (x + 1) – (x + 2) (x + 2) = -8
(v) x² – √x + 4 = 0
Solution:
(i) x² = 64 can be written as x² – 64 = 0
Which is in the form of ax² + bx + c = 0. So, it is a quadratic equation.
(ii) x² + 4x + 2 = 0 is in the form of ax² + bx + c = 0. So, it is a quadratic equation.
(iii) x² + 3/x² = 8
(x⁴ + 3) /x² = 8
x⁴ + 3 = 8x²
Which is not in the form of ax² + bx + c = 0. So, it is not a quadratic equation.
(iv) x (x + 1) – (x + 2) (x + 2) = -8
x² + x – [x² + 4x + 4] = -8
x² + x – x² – 4x – 4 = -8
-3x – 4 = -8
3x = -4 + 8 = 4
Which is not in the form of ax² + bx + c = 0. So, it is not a quadratic equation.
(v) x² – √x + 4 = 0 is not in the form of ax² + bx + c = 0. So, it is not a quadratic equation.
3. Find if the given values are the solution of the following equations:
(i) x² – 6√5x + 25 = 0; x = 5 and x = 5√5
(ii) 8x² + 2x – 1 = 0; x = 1/2 and x = -1/4
(iii) x² – 3x = 40; x = -5, x = 8
(iv) 6x² – 13x + 5 = 0; x = 1/2, and x = 5/3
Solution:
(i) Given equation is x² – 6√5x + 25 = 0
a = 1, b = -6√5, c = 25
roots (α, β) = [-b ± √(b² – 4ac)] / 2a
= [-(-6√5) ± √((-6√5)² – 4 * 1 * 25)] / (2 * 1)
= [6√5 ± √(180 – 100)] / 2
= [6√5 ± √80] / 2
= [6√5 ± 4√5] / 2
= [6√5 + 4√5] / 2 and [6√5 – 4√5] / 2
= 10√5 / 2 and 2√5 / 2
= 5√5 and √5
So, the roots are satisfying the equation.
(ii) Given quadratic equation is 8x² + 2x – 1 = 0
8x² + 4x – 2x – 1 = 0
= 4x(2x + 1) -1(2x + 1) = 0
(2x + 1)(4x – 1) = 0
2x + 1 = 0 and 4x – 1 = 0
2x = -1 and 4x = 1
x = -1/2 and x = 1/4
Therefore, the given roots are correct.
(iii) Given quadratic equation is x² – 3x = 40
It can also be written as x² – 3x – 40 = 0
By using the factorization method
x² – 8x + 5x – 40 = 0
x(x – 8) + 5(x – 8) = 0
(x + 5)(x – 8) = 0
x – 8 = 0 and x + 5 = 0
x = 8 and x = -5
Therefore, the given roots are correct.
(iv) Given details are 6x² – 13x + 5 = 0; x = 1/2, and x = 5/3
6x² – 10x – 3x + 5 = 0
2x(3x – 5) – 1(3x – 5) = 0
(2x – 1)(3x – 5) = 0
2x – 1 = 0 and 3x – 5 = 0
2x = 1 and 3x = 5
x = 1/2 and x = 5/3
Therefore, the given roots are correct.
4. Solve the following quadratic equations and find the solution.
(i) x² – (√2 – √3)x – √6 = 0
(ii) 5x² + 49x + 72 = 0
(iii) x² + 2x – 24 = 0
Solution:
(i) Given quadratic equation is x² – (√2 – √3)x – √6
a = 1, b = √3 – √2, c = -√6
Quadratic equation roots formula is given as
roots (α, β) = [-b ± √(b² – 4ac)] / 2a
= [-(√3 – √2) ± √((√3 – √2)² – 4 * 1 * (-√6)] / 2 * 1
= [(√2 – √3) ± √(5 – 2√6 + 4√6)] / 2
= [(√2 – √3) ± √(5 + 2√6)] / 2
= [(√2 – √3) + √(5 + 2√6)] / 2 and [(√2 – √3) – √(5 + 2√6)] / 2
= [1.141 – 1.732 + √(5 + 2(2.44)] / 2 and [1.141 – 1.732 – √(5 + 2(2.44)] / 2
= [-0.591 + √(5 + 4.88)] / 2 and [-0.591 – √(5 + 4.88)] / 2
= [-0.591 + √9.88] / 2 and [-0.591 – √9.88] / 2
= [-0.591 + 3.143] / 2 and [-0.591 – 3.143] / 2
= 2.55 / 2 and -3.734 / 2
= 1.275 and -1.875
= √2 and -√3
(ii) Given equation is 5x² + 49x + 72 = 0
5x² + 40x + 9x + 72 = 0
5x(x + 8) +9(x + 8) = 0
(5x + 9) (x + 8) = 0
(5x + 9) = 0 and (x + 8) = 0
5x = -9 and x = -8
x = -9/5
The roots are (-8, -9/5)
(iii) Given equation is x² + 2x – 24 = 0
x² + 6x – 4x – 24 = 0
x(x + 6) -4(x + 6) = 0
(x – 4)(x + 6) = 0
(x – 4) = 0 and (x + 6) = 0
x = 4 and x = -6
The roots are (4, -6).
5. Find the roots for the following quadratic equations:
(i) 50x² + 110x + 48 = 0
(ii) 4x² + 12x + 9 = 0
(iii) 16x² – 49 = 0
Solution:
(i) Given quadratic equation is 50x² + 110x + 48 = 0
50x² + 80x + 30x + 48 = 0
10x(5x + 8) + 6(5x + 8) = 0
(10x + 6) (5x + 8) = 0
(10x + 6) = 0 and (5x + 8) = 0
10x = -6 and 5x = -8
x = -6/10 and x = -8/5
Roots are (-3/5, -8/5)
(ii) Given quadratic equation is 4x² + 12x + 9 = 0
4x² + 6x + 6x + 9 = 0
2x(2x + 3) +3(2x + 3) = 0
(2x + 3) (2x + 3) = 0
(2x + 3) = 0 and (2x + 3) = 0
2x = -3 and 2x = -3
x = -3/2 and x = -3/2
The roots are (-3/2, -3/2)
(iii) Given quadratic equation is 16x² – 49 = 0
(4x)² – 7² = 0
(4x)² = 7²
Applying square root on both sides
√(4x)² = √7²
4x = ±7
x = 7/ 4 and x = -7/4
The roots are (-7/4, 7/4).