Discriminant: Number of Solutions, Examples

The discriminant is $b^2 - 4ac$ , the part under the square root in the quadratic formula. It tells you how many real solutions $ax^2 + bx + c = 0$ has, without solving it. If it is positive, there are two. If it is zero, there is one. If it is negative, there are none.

Before this lesson: You should know the quadratic formula, graphing quadratic functions and solving quadratics by graphing.

What Does the Discriminant Tell You?

Look at the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The number under the square root decides what kind of answers you get.

If that number is positive, the ± gives two different answers. If it is 0, adding or subtracting 0 gives the same answer twice. If it is negative, there is no real square root, so there are no real answers.

You can see this on a graph. The real solutions are where the parabola meets the x-axis.

Three parabolas side by side on grids from −3 to 3. Left: y = x² − 1 crosses the x-axis twice, at −1 and 1; its discriminant is 4, which is positive, so there are 2 real solutions. Middle: y = x² − 2x + 1 touches the x-axis once, at 1; its discriminant is 0, so there is 1 real solution. Right: y = x² + 1 never reaches the x-axis; its discriminant is −4, which is negative, so there are no real solutions. — Positive discriminant: the graph crosses the x-axis twice. Zero: it touches once. Negative: it misses the x-axis.

The Discriminant Formula

\text{Discriminant} = b^2 - 4ac

Discriminant	Number of real solutions	What the graph does
$b^2 - 4ac \gt 0$	Two	Crosses the x-axis twice
$b^2 - 4ac = 0$	One	Touches the x-axis once, at the vertex
$b^2 - 4ac \lt 0$	None	Never meets the x-axis

One more clue: when $a$ , $b$ and $c$ are integers and the discriminant is a perfect square, like 16 or 49, the solutions are rational. That means the quadratic can be factored.

Steps for using the discriminant

Write the equation in standard form: $ax^2 + bx + c = 0$ .
Find $a$ , $b$ and $c$ , with their signs.
Find $b^2 - 4ac$ .
Check if it is positive, zero or negative.

Worked Examples

Example 1: A positive discriminant

How many real solutions does $x^2 - 6x + 5 = 0$ have?

b^2 - 4ac = (-6)^2 - 4(1)(5) = 36 - 20 = 16

16 is positive, so there are two real solutions. It is also a perfect square, so the quadratic factors: $(x - 1)(x - 5) = 0$ .

Example 2: A discriminant of zero

How many real solutions does $x^2 + 4x + 4 = 0$ have?

b^2 - 4ac = 4^2 - 4(1)(4) = 16 - 16 = 0

The discriminant is 0, so there is one real solution. It is $x = -2$ , since $x^2 + 4x + 4 = (x + 2)^2$ .

Example 3: A negative discriminant

How many real solutions does $2x^2 + 3x + 4 = 0$ have?

b^2 - 4ac = 3^2 - 4(2)(4) = 9 - 32 = -23

The discriminant is negative, so there are no real solutions. The graph never meets the x-axis.

Example 4: Write it in standard form first

How many real solutions does $3x^2 = 2x + 4$ have?

Move every term to one side: $3x^2 - 2x - 4 = 0$ . So $a = 3$ , $b = -2$ and $c = -4$ .

b^2 - 4ac = (-2)^2 - 4(3)(-4) = 4 + 48 = 52

52 is positive, so there are two real solutions. 52 is not a perfect square, so the answers have square roots in them.

Example 5: Find k

For what values of $k$ does $x^2 + kx + 9 = 0$ have exactly one solution?

One solution means the discriminant is 0. Here $a = 1$ , $b = k$ and $c = 9$ .

\begin{aligned} k^2 - 4(1)(9) &= 0 \\ k^2 &= 36 \\ k &= \pm 6 \end{aligned}

Check: $x^2 + 6x + 9 = (x + 3)^2$ and $x^2 - 6x + 9 = (x - 3)^2$ . Each has one solution.

Example 6: Word problem

A ball is tossed up. Its height after $t$ seconds is $h = -16t^2 + 40t + 4$ feet. Does it ever reach 30 feet?

Set $h = 30$ and write the equation in standard form: $-16t^2 + 40t - 26 = 0$ .

b^2 - 4ac = 40^2 - 4(-16)(-26) = 1600 - 1664 = -64

The discriminant is negative, so there is no time when the height is 30 feet. The ball never gets that high. (Its highest point is 29 feet.)

Common Mistakes

Mistake	Example of the error	Fix
Losing the sign of c	For $3x^2 - 2x - 4$ , writing $4 - 48$	$-4(3)(-4) = +48$ , so it is $4 + 48 = 52$ .
Squaring a negative b wrong	$(-6)^2 = -36$	$(-6)^2 = 36$ . The square is never negative.
Taking the square root	Discriminant $= \sqrt{b^2 - 4ac}$	The discriminant is just $b^2 - 4ac$ . No square root.
Not in standard form	For $3x^2 = 2x + 4$ , using $b = 2$	Move every term to one side first, so $b = -2$ and $c = -4$ .
Thinking 0 means no solution	$b^2 - 4ac = 0$ , so no answer	Zero means exactly one real solution. Only a negative discriminant means none.

Practice Problems

How many real solutions does $x^2 + 5x + 6 = 0$ have?
How many real solutions does $x^2 - 10x + 25 = 0$ have?
How many real solutions does $x^2 + x + 3 = 0$ have?
How many real solutions does $2x^2 - 5x - 1 = 0$ have?
How many real solutions does $4x^2 + 12x + 9 = 0$ have?
How many real solutions does $3x^2 - x + 2 = 0$ have?
Find the discriminant of $x^2 - 7x = -10$ . Can the quadratic be factored?
Find $k$ so that $x^2 - 8x + k = 0$ has exactly one solution.
For what values of $k$ does $x^2 + 6x + k = 0$ have two real solutions?
A rock is thrown up. Its height is $h = -16t^2 + 48t + 5$ feet after $t$ seconds. Does it reach 45 feet?

Show answers

$5^2 - 4(1)(6) = 25 - 24 = 1$ . Positive, so two real solutions.
$(-10)^2 - 4(1)(25) = 100 - 100 = 0$ . One real solution.
$1^2 - 4(1)(3) = 1 - 12 = -11$ . Negative, so no real solutions.
$(-5)^2 - 4(2)(-1) = 25 + 8 = 33$ . Positive, so two real solutions.
$12^2 - 4(4)(9) = 144 - 144 = 0$ . One real solution.
$(-1)^2 - 4(3)(2) = 1 - 24 = -23$ . Negative, so no real solutions.
Standard form: $x^2 - 7x + 10 = 0$ . $(-7)^2 - 4(1)(10) = 49 - 40 = 9$ . 9 is a perfect square, so yes: $(x - 2)(x - 5) = 0$ .
$(-8)^2 - 4(1)(k) = 0$ , so $64 - 4k = 0$ and $k = 16$ .
$6^2 - 4(1)(k) \gt 0$ , so $36 - 4k \gt 0$ and $k \lt 9$ .
Set $h = 45$ : $-16t^2 + 48t - 40 = 0$ . $48^2 - 4(-16)(-40) = 2304 - 2560 = -256$ . Negative, so no. The rock never reaches 45 feet.

FAQ

What is the discriminant in math?

It is $b^2 - 4ac$ for a quadratic $ax^2 + bx + c = 0$ . It tells you how many real solutions the equation has.

What does a negative discriminant mean?

It means the equation has no real solutions. The parabola never touches the x-axis.

What does it mean when the discriminant is 0?

There is exactly one real solution. The vertex of the parabola sits right on the x-axis.

How does the discriminant tell you if a quadratic can be factored?

If $a$ , $b$ and $c$ are integers and the discriminant is a perfect square, the quadratic factors over the integers. If not, use the quadratic formula.

Summary

The discriminant is $b^2 - 4ac$ .
Positive: two real solutions. Zero: one. Negative: none.
The graph crosses the x-axis twice, touches it once or misses it.
A perfect-square discriminant means the quadratic can be factored.
Always write the equation in standard form first.

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