18.2 How Many Roots Can a Quadratic Graph Have?

🟦 Concept

A root of a quadratic graph is an $x$-value where the curve meets the x-axis.

At a root:$$y=0$$

A quadratic graph can have:

• Two real roots
• One repeated real root
• No real roots

The number of roots depends on how the parabola meets the x-axis.

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🟩 1. Two Real Roots

If the quadratic curve crosses the x-axis at two different points, it has two real roots.

For example:$$y=x^2-4x+3$$

Factorising:$$y=(x-1)(x-3)$$

The roots are:$$x=1$$

and$$x=3$$

Therefore, the x-intercepts are:$$(1,0)$$

and$$(3,0)$$

Graphically

The parabola crosses the x-axis twice.

Two crossings → Two real roots

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🟪 2. One Repeated Root

A quadratic has one repeated root when the curve touches the x-axis at exactly one point and then turns back.

For example:$$y=(x-2)^2$$

The root is:$$x=2$$

The graph touches the x-axis at:$$(2,0)$$

This point is also the turning point.

Graphically

The parabola touches the x-axis once but does not cross it.

One touch → One repeated root

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🟨 3. No Real Roots

A quadratic has no real roots when the curve does not meet the x-axis at all.

For example:$$y=x^2+2$$

The minimum value of $y$ is:$$2$$

Since the graph always stays above the x-axis, there are no x-intercepts.

Graphically

The parabola never touches or crosses the x-axis.

No intersection → No real roots

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🟧 Using the Discriminant

For a quadratic equation:$$ax^2+bx+c=0$$

the discriminant is:$$b^2-4ac$$

It can also tell us how many real roots the quadratic has.

If:$$b^2-4ac>0$$

there are two distinct real roots.

If:$$b^2-4ac=0$$

there is one repeated real root.

If:$$b^2-4ac<0$$

there are no real roots.

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💡 Exam Tip

When a graph is given, look at the x-axis first.

Count how many times the parabola meets it:

Crosses twice → 2 roots

Touches once → 1 repeated root

Does not meet it → 0 real roots

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⚠️ Common Mistake

Do not count the y-intercept as a root.

A root must lie on the x-axis, where:$$y=0$$

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⭐ Quick Summary

A quadratic graph can have:

Graph behaviour	Number of real roots
Crosses the x-axis twice	2
Touches the x-axis once	1 repeated root
Does not meet the x-axis	0

Remember

$$\text{Roots are where the quadratic graph meets the x-axis.}$$

A quadratic can have at most two real roots.