Math · College algebra · Concept

The quadratic formula and the discriminant

The quadratic formula x = (−b ± √(b² − 4ac))/(2a) solves any equation ax² + bx + c = 0 with a ≠ 0. The discriminant b² − 4ac tells you, before you solve, whether there are two, one or no real solutions.

When to use the formula

Write the equation in standard form, ax² + bx + c = 0, and read off a, b and c with their signs. If the quadratic factors over the integers, factoring is often quicker. The formula always works, including when the solutions are irrational or complex.

a x^{2} + b x + c = 0 (a \neq 0) ⟹ x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

\begin{array}{c} a x^{2} + b x + c = 0 (a \neq 0) \\ x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{array}

Where the formula comes from

Completing the square on the general equation produces the formula. Divide by a, move the constant term, add (b/2a)² to both sides, then take both square roots. The ± is there because a positive number has two square roots.

\begin{aligned} x^{2} + \frac{b}{a} x & = - \frac{c}{a} \\ {(x + \frac{b}{2 a})}^{2} & = \frac{b^{2} - 4 a c}{4 a^{2}} \\ x + \frac{b}{2 a} & = \pm \frac{\sqrt{b^{2} - 4 a c}}{2 a} \end{aligned}

Why is the denominator 2a and not 2|a|?

The square root of 4a² is 2|a|, but the ± already includes both signs, so writing 2a gives the same pair of solutions.

The discriminant counts the real solutions

The quantity under the root, D = b² − 4ac, is the discriminant. A positive discriminant gives two different real solutions. Zero gives one repeated solution, x = −b/(2a). A negative discriminant gives no real solutions, because a negative number has no real square root; the two solutions are then complex conjugates.

D = b^{2} - 4 a c

What the discriminant tells you
Discriminant	Real solutions	Graph of y = ax² + bx + c
D > 0	Two	Crosses the x-axis twice
D = 0	One (repeated)	Touches the x-axis at its vertex
D < 0	None (two complex)	Stays above or below the x-axis

Solutions are x-intercepts

The real solutions of ax² + bx + c = 0 are the x-intercepts of the parabola y = ax² + bx + c, because those are the points where y = 0. A graph is a good check on how many solutions there are and roughly where, while the formula gives their exact values.

Keep answers exact, then round

Simplify the radical and keep the exact form, such as (−3 ± √17)/4. Round only when a decimal is asked for, and round each solution separately. Substituting a rounded value leaves a small nonzero remainder; that is rounding, not an error.

Common mistakes

Using the formula before the equation is in standard form: in x² = 3x + 4, move every term to one side first, so b = −3 and c = −4.
Squaring a negative b incorrectly: with b = −3, b² is (−3)² = 9, not −9.
Dividing only the square root by 2a: the whole numerator −b ± √(b² − 4ac) is over 2a.
Calling a negative discriminant “no solution” instead of “no real solution”: the equation has two complex solutions.

Key terms

Quadratic formula: The solutions of ax² + bx + c = 0 with a ≠ 0: x = (−b ± √(b² − 4ac))/(2a). It comes from completing the square and works for every quadratic, including those that do not factor over the integers.
Discriminant: The quantity b² − 4ac under the root in the quadratic formula. A positive value gives two real solutions, zero gives one repeated real solution, and a negative value gives no real solutions (two complex ones).
Completing the square: Rewriting x² + bx as (x + b/2)² − (b/2)² by adding and subtracting (b/2)². It turns a quadratic equation into a square equal to a constant, and a quadratic function into vertex form.
x-intercept: A point where a graph crosses or touches the x-axis, so its y-value is 0. For y = f(x), its x-value is a zero (root) of f.
Zero-product property: If a product equals zero, at least one of its factors is zero. That is why you move everything to one side, leaving 0, before factoring to solve.
Complex number: A number a + bi, where a and b are real and i² = −1. It can be plotted as the point (a, b) in a plane; real numbers are the ones with b = 0.

Work through an example

Solve 2x² + 3x − 1 = 0. Give the exact solutions and their values to three decimal places.

Solve a quadratic with the quadratic formula →

Solve a quadratic by completing the square →

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