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Solving and using quadratic equations - Higher

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# Solving a linear and quadratic simultaneous equation

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

This is the graph of: y = x2 - 9x + 20

We can use it to solve: x2 - 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 = x2 - 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:

x2 - 9x + 20 = x - 1

One side of this equation is x2 - 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 x2 - 9x + 20 = x - 1 can be found where the line and curve crosses.
• 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

1. 32 - 9 x 3 + 20 = 3 - 1
2. 9 - 27 + 20 = 2
3. 9 - 7 = 2
4. 2 = 2

x = 7

1. 72 - 9 x 7 + 20 = 7 - 1
2. 49 - 63 + 20 = 6
3. 49 - 43 = 6
4. 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: x2 + x + 2 = 5 – x

First draw the curve y = x2 + x + 2 and the line 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: x2 + 2x + 1 = x + 4

First draw the curve y = x2 + 2x + 1 and the line: 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

• -2.32 + (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

• -1.32 + (2 x 1.3) + 1 = 1.3 + 4
• 1.69 + 2.6 + 1 = 5.3
• 5.29 and 5.3 are very close

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