The Discriminant & Nature of Roots

Q: Does a negative discriminant mean the equation has no solutions?

It has no real solutions — the parabola never touches the $x$-axis. But in Algebra 2, the equation still has two complex solutions of the form $a \pm bi$. Whether you answer 'no real solutions' or 'two complex solutions' depends on which number system the question is working in.

Q: Why is there only one solution when the discriminant is zero?

The quadratic formula becomes $x = \dfrac{-b \pm \sqrt{0}}{2a}$, and adding or subtracting zero gives the same number. Graphically, the vertex of the parabola sits exactly on the $x$-axis, so it touches at one point instead of crossing at two. Some books call this a 'double root' or a root with multiplicity two.

Q: Do I need the whole quadratic formula to find the discriminant?

No — the discriminant is only the part under the square root, $b^2 - 4ac$. You compute just that one number. If the question then asks for the actual solutions, you already have the hardest piece of the formula done.

Before you grind through the quadratic formula, there's a shortcut that tells you what kind of answer you're going to get. The discriminant is the piece under the square root in the quadratic formula: $b^2 - 4ac$ . Its sign — positive, zero, or negative — tells you how many solutions the equation $ax^2 + bx + c = 0$ has, and whether they're real or complex.

That's useful for more than saving time. Test questions constantly ask 'how many real solutions does this equation have?' or 'describe the nature of the roots' — and the discriminant answers both in one calculation, no solving required.

The three cases

Compute $b^2 - 4ac$ and check its sign. If it's positive, the equation has two different real solutions. If it's exactly zero, there's one real solution (a repeated root). If it's negative, there are no real solutions — instead there are two complex solutions involving $i$ .

Why the sign decides everything: the quadratic formula is $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The $\pm$ in front of the square root is what creates two answers. A positive number under the root gives two different values; zero under the root makes the $\pm$ meaningless, leaving one value; a negative under the root forces imaginary numbers into the answer.

There's a graphing version of the same story. The solutions of $ax^2 + bx + c = 0$ are the $x$ -intercepts of the parabola $y = ax^2 + bx + c$ . Positive discriminant: the parabola crosses the $x$ -axis twice. Zero: it touches the axis at exactly one point (the vertex sits on the axis). Negative: it never reaches the axis at all.

The three parabolas below show all three cases at once: the lower one crosses the $x$ -axis twice (positive discriminant), the middle one just touches it (zero), and the top one never reaches it (negative).

How to compute it without sign errors

First write the equation in standard form $ax^2 + bx + c = 0$ and read off $a$ , $b$ , and $c$ with their signs. Then substitute into $b^2 - 4ac$ using parentheses everywhere.

Two traps cause almost every wrong answer. First, $b^2$ means the square of the whole value of $b$ : if $b = -6$ , then $b^2 = (-6)^2 = 36$ , never $-36$ . Second, watch the sign of $c$ : in $-4ac$ , a negative $c$ makes the whole term positive, so the discriminant grows.

A bonus read: rational or irrational?

When the discriminant is positive, its exact value tells you one more thing. If it's a perfect square (like $25$ or $49$ ), the square root comes out clean, the two solutions are rational, and the quadratic factors over the integers. If it's positive but not a perfect square (like $13$ ), the solutions are real but irrational — the quadratic formula or completing the square is your only route, because factoring won't work.

Worked examples

Example 1: positive discriminant, two real solutions

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

Identify the coefficients

a = 1, \quad b = 5, \quad c = 6

Substitute into

b^2 - 4ac

5^2 - 4(1)(6)

Simplify

25 - 24 = 1

The discriminant is positive, so there are two real solutions

Answer: Two real solutions (the discriminant is $1 > 0$ )

Example 2: zero discriminant, one real solution

How many solutions does $4x^2 + 12x + 9 = 0$ have?

Identify the coefficients

a = 4, \quad b = 12, \quad c = 9

Substitute into

b^2 - 4ac

12^2 - 4(4)(9)

Simplify

144 - 144 = 0

A zero discriminant means one repeated real solution — this quadratic is the perfect square

(2x + 3)^2

Answer: One real solution (the discriminant is $0$ )

Example 3: negative discriminant, two complex solutions

Describe the solutions of $2x^2 - 3x + 5 = 0$ .

Identify the coefficients — note

b

is negative

a = 2, \quad b = -3, \quad c = 5

Substitute into

b^2 - 4ac

, squaring all of

b

(-3)^2 - 4(2)(5)

Simplify

9 - 40 = -31

The discriminant is negative, so the solutions are complex

Answer: Two complex solutions (the discriminant is $-31 < 0$ )

Example 4: a perfect-square discriminant means it factors

Will $2x^2 + 7x + 3 = 0$ factor, or do you need the quadratic formula?

Identify the coefficients

a = 2, \quad b = 7, \quad c = 3

Substitute into

b^2 - 4ac

7^2 - 4(2)(3)

Simplify

49 - 24 = 25

25

is a perfect square, so the roots are rational and the quadratic factors:

(2x + 1)(x + 3) = 0

Answer: It factors — the discriminant $25$ is a perfect square, giving $x = -\dfrac{1}{2}$ or $x = -3$

Try one yourself

Common questions

Does a negative discriminant mean the equation has no solutions?

It has no real solutions — the parabola never touches the $x$ -axis. But in Algebra 2, the equation still has two complex solutions of the form $a \pm bi$ . Whether you answer 'no real solutions' or 'two complex solutions' depends on which number system the question is working in.

Why is there only one solution when the discriminant is zero?

The quadratic formula becomes $x = \dfrac{-b \pm \sqrt{0}}{2a}$ , and adding or subtracting zero gives the same number. Graphically, the vertex of the parabola sits exactly on the $x$ -axis, so it touches at one point instead of crossing at two. Some books call this a 'double root' or a root with multiplicity two.

Do I need the whole quadratic formula to find the discriminant?

No — the discriminant is only the part under the square root, $b^2 - 4ac$ . You compute just that one number. If the question then asks for the actual solutions, you already have the hardest piece of the formula done.

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The Discriminant & Nature of Roots

The three cases

How to compute it without sign errors

A bonus read: rational or irrational?

Worked examples

Example 1: positive discriminant, two real solutions

Example 2: zero discriminant, one real solution

Example 3: negative discriminant, two complex solutions

Example 4: a perfect-square discriminant means it factors

Try one yourself

Common questions

Keep learning

Quadratic Formula

The Imaginary Unit i & Powers of i

Axis of Symmetry