We can solve a quadratic equation by factoring, completing the square, using the quadratic formula or using the graphical method.
Compared to the other methods, the graphical method only gives an estimate to the solution(s). If the graph of the quadratic function crosses the x-axis at two points then we have two solutions. If the graph touches the x-axis at one point then we have one solution. If the graph does not intersect with the x-axis then the equation has no real solution.
The following video explains how the quadratic graph can show the number of solutions for the quadratic equation and the values of the solutions.We will now graph a quadratic equation that has two solutions. The solutions are given by the two points where the graph intersects the x-axis.
Example:
Solve the equation x^{2} + x – 3 = 0 by drawing its graph for –3 ≤ x ≤ 2.
Solution:
Rewrite the quadratic equation x^{2} + x – 3 = 0 as the quadratic function y = x^{2} + x – 3
Draw the graph for y = x^{2} + x – 3 for –3 ≤ x ≤ 2.
x |
–3 |
–2 |
–1 |
0 |
1 |
2 |
y |
3 |
–1 |
–3 |
–3 |
–1 |
3 |
The solution for the equation x^{2} + x – 3 can be obtained by looking at the points where the graph y = x^{2} + x – 3 cuts the x-axis (i.e. y = 0).
The graph y = x^{2} + x – 3, cuts the x-axis at x 1.3 and x –2.3
So, the solution for the equation x + x –3 is x 1.3 or x –2.3.
Recall that in the quadratic formula, the discriminant b^{2} – 4ac is positive when there are two distinct real solutions (or roots).Example:
By plotting the graph, solve the equation 6x – 9 – x^{2} = 0.
Solution:
x |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
y |
–9 |
–4 |
–1 |
0 |
–1 |
–4 |
–9 |
Notice that the graph does not cross the x-axis, but touches the x-axis at x = 3. This means that the equation 6x – 9 – x^{2} = 0 has one solution (or equal roots) of x = 3.
Recall that in the quadratic formula, in such a case where the roots are equal, the discriminant b^{2} – 4ac = 0.Example:
Solve the equation x^{2} + 4x + 8 = 0 using the graphical method.
Solution:
x |
–4 |
–3 |
–2 |
–1 |
0 |
1 |
y |
8 |
5 |
4 |
5 |
8 |
13 |
Notice that the graph does not cross or touch the x-axis. This means that the equation x^{2} + 4x + 8 = 0 does not have any real solution (or roots).
Recall that in the quadratic formula, the discriminant b^{2} – 4ac, is negative when there are no real solution (or roots).
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