What if the discriminant is zero?

The equation has exactly one real root (a repeated root), evaluated as $x = \frac{-b}{2a}$.

What if the discriminant is negative?

The roots are complex numbers. You must evaluate the square root of the negative discriminant using the imaginary unit $i = \sqrt{-1}$.

What if b or c is zero?

The quadratic formula still yields the correct roots. However, factoring or isolating $x^2$ is computationally faster.

How to use the quadratic formula

The quadratic formula finds the roots of a quadratic equation by evaluating $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . It applies to any polynomial equation of degree two that is expressed in the standard form $ax^2 + bx + c = 0$ .

The setup

Rearrange the equation into standard form $ax^2 + bx + c = 0$ . Identify the constant coefficients $a$ , $b$ , and $c$ . Ensure that $a eq 0$ , otherwise the equation is linear, not quadratic.

The steps

Calculate the discriminant $\Delta = b^2 - 4ac$ . 2. Substitute $a$ , $b$ , and $\Delta$ into the quadratic formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ . 3. Evaluate the positive and negative cases to find the two roots.

Checking the result

Substitute the calculated values of $x$ back into the original equation $ax^2 + bx + c = 0$ . The left side of the equation must evaluate exactly to zero.

Common errors

Failing to put the equation in standard form before identifying coefficients is a primary source of error. Other common mistakes include dropping negative signs when squaring $b$ or evaluating $-4ac$ , and incorrectly canceling terms across the numerator and denominator.

Worked example

Find the roots of $2x^2 - 5x = 3$ .

Standard form: $2x^2 - 5x - 3 = 0$ . Identify variables: $a = 2$ , $b = -5$ , $c = -3$ . Calculate discriminant: $\Delta = (-5)^2 - 4(2)(-3) = 25 + 24 = 49$ . Apply formula: $x = \frac{-(-5) \pm \sqrt{49}}{2(2)}$ . Simplify numerator and denominator: $x = \frac{5 \pm 7}{4}$ . Calculate first root: $x_1 = \frac{12}{4} = 3$ . Calculate second root: $x_2 = \frac{-2}{4} = -0.5$ .

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References: OpenStax College Algebra, Chapter 2.5 · Khan Academy, Quadratic equations