Quadratic equations

A quadratic equation contains up to \(x^2\). There are many ways to solve quadratics. All quadratic equations can be written in the form \(ax^2 + bx + c = 0\) where \(a\), \(b\) and \(c\) are numbers (\(a\) cannot be equal to 0, but \(b\) and \(c\) can be 0).

Here are some examples of quadratic equations in this form:

\(2x^2 - 2x - 3 = 0\). \(a = 2\), \(b = -2\) and \(c = -3\)
\(2x(x + 3) = 0\). \(a = 2\), \(b = 6\) and \(c = 0\) (in this example, the bracket can be expanded to \(2x^2 + 6x = 0\))
\((2x + 1)(x - 5) = 0\). \(a = 2\), \(b = -9\) and \(c = -5\) (this will expand to \(2x^2 - 9x - 5 = 0\))
\(x^2 + 2 = 4\). \(a = 1\), \(b = 0\) and \(c = -2\) (this equation rearranges to \(x^2 – 2 = 0\)
\(3x^2 = 48\). \(a = 3\), \(b = 0\) and \(c = -48\) (this equation rearranges to \(3x^2 - 48 = 0\))

Solving a quadratic equation by graph

If the graph \(y = ax^2 + bx + c \) crosses the x-axis, the values of \(x\) at the crossing points are the roots or solutions of the equation \(ax^2 + bx + c = 0 \).

If the equation \(ax^2 + bx + c = 0 \) has just one solution (a repeated root) then the graph just touches the x-axis without crossing it. If the equation \(ax^2 + bx + c = 0 \) has no solutions then the graph does not cross or touch the x-axis.

Example

Solve \(x^2 – x – 4 = 0\) by graph, giving your answers to 1 decimal place.

Start with a table of values to find coordinates of points on the graph.

x	-3	-2	-1	0	1	2	3	4	5
y	8	2	-2	-4	-4	-2	2	8	16

x
-3
-2
-1
0
1
2
3
4
5

y
8
2
-2
-4
-4
-2
2
8
16

Plot these points and join them with a smooth curve.

The roots of the equation y = x^2 -x – 4 are the x-coordinates where the graph crosses the x-axis, which can be read from the graph: x = -1.6 and x=2.6 (1 dp)

The roots of the equation \( x^2 -x – 4 = 0\) are the x-coordinates where the graph crosses the x-axis, which can be read from the graph:\(x = -1.6 \) and \(x = 2.6 \) (1dp).

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