# Quadratic Equations from Algebraic Fractions

In this worksheet, students add or subtract algebraic fractions and simplify to form quadratic equations.

Key stage:  KS 4

Curriculum topic:  Algebra

Curriculum subtopic:  Solve Quadratic Equations by Factorising

Difficulty level:

### QUESTION 1 of 10

When we add or subtract two fractions, they must have the same denominator.

We can multiply a complete equation by a lowest common denominator to "get rid of" all denominators, making the equation easier to solve.

Example

Simplify the following equation to one that is quadratic:

 2 + 3 = 4 x - 1 x + 2

We multiply the whole equation by (x - 1) (x + 2).

This gives us:

2 (x + 2) + 3 (x - 1) = 4 (x - 1) (x + 2)

Simplify to get:

2x + 4 + 3x - 3 = 4 (x2 + x - 2)

5x + 1 = 4x2 + 4x - 8

4x2 + 4x - 8 - 5x - 1 = 0

4x2 - x - 9 = 0

This can now be solved using the quadratic formula, but for this worksheet we just need to arrive at this quadratic equation in the form ax2 + bx + c = 0

Simplify the following equation to one that is quadratic:

 2 + 3 = 4 x + 1 x + 2
4x² + 3x + 6 = 0

4x² + 2x + 6 = 0

4x² + 7x + 1 = 0

Simplify the following equation to one that is quadratic:

 2 + 3 = 1 x + 1 x + 3
x² - x - 6 = 0

x² + 2x - 6 = 0

x² + 6x - 2 = 0

Simplify the following equation to one that is quadratic:

 2 - 3 = 5 x + 3 x + 1
5x² - 22x - 21 = 0

5x² + 22x - 21 = 0

5x² + 21x + 22 = 0

Simplify the following equation to one that is quadratic:

 3 - 1 = 2 x + 3 x - 1
2x² - 2x - 2 = 0

2x² + 2x = 0

2x² + 2x + 2 = 0

Simplify the following equation to one that is quadratic:

 3 - 1 = 2 x - 3 x + 1
2x² - 2x - 12 = 0

x² - 3x - 6 = 0

2x² - 6x + 12 = 0

Simplify the following equation to one that is quadratic:

 4 + 2 = 3 x - 2 x + 5
3x² + 3x - 14 = 0

x² - 3x - 15 = 0

3x² + 3x - 46 = 0

Simplify the following equation to one that is quadratic:

 1 + 2 = 2 x - 4 x + 3
2x² - 5x - 19 = 0

2x² - x - 24 = 0

2x² + x - 29 = 0

Simplify the following equation to one that is quadratic:

 4 - 10 = -1 x - 4 x - 3
x² + 13x - 16 = 0

x² - x - 16 = 0

x² - 13x + 40 = 0

Simplify the following equation to one that is quadratic:

 2 + 1 = 7 2x - 1 4x + 3
56x² + 4x - 26 = 0

56x² - 24x - 10 = 0

56x² - 4x + 10 = 0

Simplify the following equation to one that is quadratic:

 2 - 1 = 4 2x - 1 3x + 1
24x² + 8x - 1 = 0

24x² - 8x - 7 = 0

24x² - 4x + 1 = 0

• Question 1

Simplify the following equation to one that is quadratic:

 2 + 3 = 4 x + 1 x + 2
4x² + 7x + 1 = 0
EDDIE SAYS
2(x + 2) + 3(x + 1) = 4(x² + 3x + 2)
2x + 4 + 3x + 3 = 4x² + 12x + 8
• Question 2

Simplify the following equation to one that is quadratic:

 2 + 3 = 1 x + 1 x + 3
x² - x - 6 = 0
EDDIE SAYS
2(x + 3) + 3(x + 1) = 1(x² + 4x + 3)
2x + 6 + 3x + 3 = x² + 4x + 3
• Question 3

Simplify the following equation to one that is quadratic:

 2 - 3 = 5 x + 3 x + 1
5x² + 21x + 22 = 0
EDDIE SAYS
2(x + 1) - 3(x + 3) = 5(x² + 4x + 3)
2x + 2 - 3x - 9 = 5x² + 20x + 15
• Question 4

Simplify the following equation to one that is quadratic:

 3 - 1 = 2 x + 3 x - 1
2x² + 2x = 0
EDDIE SAYS
3(x - 1) - 1(x + 3) = 2(x² + 2x - 3)
3x - 3 - x - 3 = 2x² + 4x - 6
• Question 5

Simplify the following equation to one that is quadratic:

 3 - 1 = 2 x - 3 x + 1
x² - 3x - 6 = 0
EDDIE SAYS
3(x + 1) - 1(x - 3) = 2(x² - 2x - 3)
3x + 1 - x + 3 = 2x² - 4x - 6
• Question 6

Simplify the following equation to one that is quadratic:

 4 + 2 = 3 x - 2 x + 5
3x² + 3x - 46 = 0
EDDIE SAYS
4(x + 5) + 2(x - 2) = 3(x² + 3x - 10)
4x + 20 + 2x - 4 = 3x² + 9x - 30
• Question 7

Simplify the following equation to one that is quadratic:

 1 + 2 = 2 x - 4 x + 3
2x² - 5x - 19 = 0
EDDIE SAYS
1(x + 3) + 2(x - 4) = 2(x² - x - 12)
x + 3 + 2x - 8 = 2x² - 2x - 24
• Question 8

Simplify the following equation to one that is quadratic:

 4 - 10 = -1 x - 4 x - 3
x² - 13x + 40 = 0
EDDIE SAYS
4(x - 3) - 10(x - 4) = -1(x² - 7x + 12)
4x - 12 - 10x + 40 = -x² + 7x - 12
• Question 9

Simplify the following equation to one that is quadratic:

 2 + 1 = 7 2x - 1 4x + 3
56x² + 4x - 26 = 0
EDDIE SAYS
2(4x + 3) + 1(2x - 1) = 7(8x² + 2x - 3)
8x + 6 + 2x - 1 = 56x² + 14x - 21
• Question 10

Simplify the following equation to one that is quadratic:

 2 - 1 = 4 2x - 1 3x + 1