Quadratic Formula
In short: The quadratic formula solves any equation ax² + bx + c = 0 with x = (−b ± √(b² − 4ac)) / (2a), giving both solutions at once.
Algebra 1AlgebraA-REI.B.4b
What is Quadratic Formula?
Some quadratics refuse to factor neatly, and the quadratic formula is the tool that always works. It reads the three coefficients of \(ax^2 + bx + c = 0\) and returns every solution.
The formula is \(x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\). The plus-or-minus sign yields two answers. The part under the root, \(b^2 – 4ac\), is the discriminant: it tells whether there are two real solutions, one, or none before you finish the arithmetic.
How to spot it
- An equation has an x-squared term and cannot be factored by inspection.
- A real situation, like a projectile hitting the ground, sets a quadratic equal to zero.
- The directions say use the quadratic formula or give exact answers with radicals.
- Coefficients are large, fractional or negative, which makes guessing factors impractical.
The method
- Move everything to one side so the equation reads \(ax^2 + bx + c = 0\), and identify a, b and c with their signs.
- Compute the discriminant \(b^2 – 4ac\) first.
- Substitute into \(x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\) and simplify the numerator and the denominator.
- Work out the plus case and the minus case separately, then check one answer in the original equation.
Quadratic Formula: solved example
Problem
A ball is tossed from a 40-foot deck so that its height is \(h = -16t^2 + 24t + 40\) feet after t seconds. When does it hit the ground?
Solution
Set \(h = 0\): \(-16t^2 + 24t + 40 = 0\). Divide by \(-8\): \(2t^2 – 3t – 5 = 0\), so \(a = 2\), \(b = -3\), \(c = -5\).
The discriminant is \(9 + 40 = 49\), so \(t = \frac{3 \pm 7}{4}\), giving \(t = 2.5\) or \(t = -1\). Time cannot be negative, so the ball lands after 2.5 seconds.
Try it
Question 1
Solve \(x^2 – 6x + 4 = 0\) and leave the answer exact.
Show the answer
\(x = \frac{6 \pm \sqrt{36 – 16}}{2} = \frac{6 \pm 2\sqrt{5}}{2} = 3 \pm \sqrt{5}\).
Question 2
How many real solutions does \(x^2 + 2x + 5 = 0\) have?
Show the answer
\(4 – 20 = -16\) is negative, so there are no real solutions.
Quadratic Formula vs. Completing the Square
The formula is what you get when you complete the square on the general equation once and for all. Completing the square does the work on each specific equation; the formula lets you skip straight to plugging in a, b and c. Use whichever is faster for the problem in front of you.
Keep going
Lessons
Worksheets and quizzes
Related words
- Discriminant
- Completing the Square
- Standard Form of a Quadratic
- Factoring
- Zero Product Property
- Quadratic Function
Whole dictionary: A–Z · Algebra 1 · Algebra