how to find the roots of an equation

Solving quadratic equations

Solve quadratic equations by factorising, using formulae and completing the square. Each method also provides information about the corresponding quadratic graph.

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Graphs of quadratic functions

All quadratic functions have the same type of curved graphs with a line of symmetry.

The graph of the quadratic function \(y = ax^2 + bx + c\) is a smooth curve with one turning point. The turning point lies on the line of symmetry.

Graph of y = ax ² + bx + c

A graph showing the turning point when a > 0 and turning point when a < 0. The turning point lies on the line of symmetry.

Finding points of intersection

Roots of a quadratic equation ax ² + bx + c = 0

If the graph of the quadratic function \(y = ax^2 + bx + c \) 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.

Finding roots graphically

When the graph of \(y = ax^2 + bx + c \) is drawn, the solutions to the equation are the values of the x-coordinates of the points where the graph crosses the \(x\) -axis.

Example

Draw the graph of \(y = x^2 -x – 4 \) and use it to find the roots of the equation to 1 decimal place.

Draw and complete 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

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).

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