## Math 154b Solving Using The Quadratic Formula Worksheet

### Worksheet on Quadratic Equations | Factoring and Solving Quadratic Equations Worksheet

Worksheet on Quadratic Equations is equipped here. Students can observe a couple of questions on discovering the roots of a quadratic equation and test whether or not the given equation is quadratic or not. The perfect tactics to resolve the quadratic equation is by means of factorization, using the formula, and by means of drawing a graph. Practice extra questions on the Quadratic Equations Worksheet without spending a dime and acquire wisdom on this concept.

We are suggesting you to refer the Quadratic Formula Worksheet with Answers and start preparation. Mostly use the factorization technique to get the roots of the equation simply.

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 type 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's not a quadratic equation.

(iii) x² = Zero 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, this can be a quadratic equation.

(v) x² – 25 = Zero is in the type 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 = Sixty four will also be written as x² – 64 = 0

Which is in the type of ax² + bx + c = 0. So, it is a quadratic equation.

(ii) x² + 4x + 2 = Zero is in the type of ax² + bx + c = 0. So, this is a quadratic equation.

(iii) x² + 3/x² = 8

(x⁴ + 3) /x² = 8

x⁴ + 3 = 8x²

Which is not in the type of ax² + bx + c = 0. So, it isn't 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 = Zero isn't in the form of ax² + bx + c = 0. So, it is not a quadratic equation.

3. Find if the given values are the answer 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 a couple of√5 / 2

= 5√5 and √5

So, the roots are pleasurable 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 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 = Zero and x + 5 = 0

x = 8 and x = -5

Therefore, the given roots are right kind.

(iv) Given main points 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 to 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) = Zero and (x + 6) = 0

x = Four 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) = Zero 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 all sides

√(4x)² = √7²

4x = ±7

x = 7/ 4 and x = -7/4

The roots are (-7/4, 7/4).