Why does the discriminant decide the number of roots?

It is the part under the square root in the quadratic formula. A positive number has two square roots, zero has one, and a negative number has no real square root.

What does the discriminant mean on a graph?

Positive: the curve crosses the x-axis twice. Zero: it touches the axis once. Negative: it never meets the axis.

How to Find the Discriminant and the Nature of Roots

Short answer

How do you find the discriminant?

For ax² + bx + c = 0, the discriminant is b² − 4ac. If it is positive there are two distinct real roots, if it is zero there is one repeated root, and if it is negative there are no real roots.

The discriminant is the part of the quadratic formula under the square root:

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Its sign tells you how many real roots there are, without solving anything.

$b^{2} - 4 a c$	Roots	Graph
$> 0$	two distinct real roots	crosses the $x$ -axis twice
$= 0$	one repeated root	touches the axis
$< 0$	no real roots	never meets the axis

Examples

$x^{2} + 5 x + 2$ : $25 - 8 = 17 > 0$ , so two distinct real roots.
$x^{2} - 6 x + 9$ : $36 - 36 = 0$ , so one repeated root.
$2 x^{2} - 3 x + 5$ : $9 - 40 = - 31 < 0$ , so no real roots.

Write the substitution with brackets, $(- 6)^{2} - 4 (1) (9)$ , so the signs cannot slip.

Using it to find an unknown

Find $k$ so that $k x^{2} + 4 x + 1 = 0$ has equal roots.

b^{2} - 4 a c = 16 - 4 k = 0 ⟹ k = 4

( $k$ cannot be 0, or the equation would not be a quadratic.)

Try it: slide k and watch the roots

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Mistakes that cost marks

Dropping the sign of $c$ . When $a$ and $c$ have opposite signs, $- 4 a c$ is positive.
Forgetting the second value. $k^{2} = 36$ gives $k = 6$ and $k = - 6$ .
Allowing $a = 0$ in a question about a quadratic.

Exam questions on this: 4037 discriminant and nature of roots.

How to find the discriminant of a quadratic