Worked examples
When asked to solve a quadratic equation, we are really finding the roots 鈥 where the parabola cuts the x-axis, therefore when we have the graph drawn, it is very easy to do this.
Example
Using the graph below, solve the quadratic equation.
Answer
Looking at the graph we can see that the roots occur at (-1, 0) and (3, 0), therefore \(x = - 1\) and \(x = 3\).
Now try the example question below.
Question
Solve the quadratic equation \({x^2} - 6x + 8\) by completing the table below, plotting the points and drawing the parabola then reading off the roots from the graph.
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 |
| \(y = {x^2} - 6x + 8\) | 8 | ... | ... | ... | ... | ... |
| \(x\) | \(y = {x^2} - 6x + 8\) |
|---|---|
| 0 | 8 |
| 1 | ... |
| 2 | ... |
| 3 | ... |
| 4 | ... |
| 5 | ... |
Substitute the values of \(x\) into the equation to get the coordinates to plot:
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 |
| \(y = {x^2} - 6x + 8\) | 8 | 3 | 0 | -1 | 0 | 3 |
| \(x\) | \(y = {x^2} - 6x + 8\) |
|---|---|
| 0 | 8 |
| 1 | 3 |
| 2 | 0 |
| 3 | -1 |
| 4 | 0 |
| 5 | 3 |
Therefore the coordinates are (0, 8), (1, 3), (2, 0), (3, -1), (4, 0) and (5, 3).
Plot these points and draw the parabola as shown below:
Therefore by reading the roots from the graph, \(x = 2\,and\,x = 4\).
More guides on this topic
- Expansion of brackets
- Factorising an algebraic expression
- Completing the square in a quadratic expression
- Simplifying algebraic fractions
- Applying the four operations to algebraic fractions
- Determining the equation of a straight line
- Working with linear equations and inequations
- Working with simultaneous equations
- Changing the subject of a formula
- Determine the equation of a quadratic function from its graph
- Sketching a quadratic function
- Identifying features of a quadratic function
- Solving a quadratic equation using the quadratic formula
- Using the discriminant to determine the number of roots
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