Another way to solve a quadratic equation is to draw its graph.

This is the graph of: **y = x ^{2} - 9x + 20**

We can use it to solve: **x ^{2} - 9x + 20 = 0**

The answers are along the x axis where the graph reaches y = 0 (where it crosses the x axis). Using the graph the answers are x = 4 and x = 5. We can also solve this equation by factorising:

- y = x
^{2}- 9x + 20 - y = (x - 4) (x - 5)

This shows that the solutions are x = 4 and x = 5 which matches the answers in the graph above.

A graphical method can be used to find the solution of a quadratic and linear equation. Suppose we want to find the solutions for:

**x ^{2} - 9x + 20 = x - 1**

One side of this equation is x^{2} - 9x + 20. We already know what this graph looks like.

The other side of the equation is x - 1. Drawing y = x - 1 on the same graph we get:

- The solutions to
**x**can be found where the line and curve crosses.^{2}- 9x + 20 = x - 1 - The solutions are found reading values from the x-axis.
- The solutions are x = 3 and x = 7.

We can check this with the following:

**x = 3**

- 3
^{2}- 9 x 3 + 20 = 3 - 1 - 9 - 27 + 20 = 2
- 9 - 7 = 2
- 2 = 2

**x = 7**

- 7
^{2}- 9 x 7 + 20 = 7 - 1 - 49 - 63 + 20 = 6
- 49 - 43 = 6
- 6 = 6

For the following questions, you may find it useful to look back at the graph sections Straight-line graphs and Curved graphs.

- Question
Solve: x

^{2}+ x + 2 = 5 – x

- Answer
First draw the curve

**y = x**and the line^{2}+ x + 2**y = 5 – x**.The solutions are where the curve and line cross. In this case x = -3 and x = 1.

The answers will not usually be whole numbers.

- Question
Solve: x

^{2}+ 2x + 1 = x + 4

- Answer
First draw the curve

**y = x**and the line:^{2}+ 2x + 1**y = x + 4**.The solutions are where the curve and line cross. In this case x = -2.3 and x =1.3.

Because these solutions are correct to one decimal place, a check will give numbers that are close but not exactly the same.

**x = -2.3**- -2.3
^{2}+ (2 x -2.3) + 1 = -2.3 + 4 - 5.29 + (-4.6) + 1 = 1.7
- 1.69 and 1.7 are very close

**x = 1.3**- -1.3
^{2}+ (2 x 1.3) + 1 = 1.3 + 4 - 1.69 + 2.6 + 1 = 5.3
- 5.29 and 5.3 are very close

- -2.3

**Now try a **Test Bite.

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