The quadratic formula gives the solutions of any quadratic equation ax2 + bx + c = 0 with a ≠ 0: x = −b ± √b2 − 4ac2a. Substitute a, b and c, then compute once with plus and once with minus. It works even when the quadratic cannot be factored.
In Spanish
fórmula cuadrática
Also called
quadratic equation formula, x = -b plus or minus formula, general formula
Each piece has a job. The discriminant decides how many answers you get; the ± produces them.
Formal definition and notation
For real numbersa ≠ 0, b and c, the solutions of ax2 + bx + c = 0 are x = −b ± √b2 − 4ac2a. The formula follows from completing the square on the general equation. The discriminantb2 − 4ac determines whether the solutions are two distinct reals, one repeated real, or two complex conjugates.
Symbol
Read it as
Meaning
±
plus or minus
two calculations: one adding the square root, one subtracting it
Solve 3x2 + 5x − 1 = 0. Give exact answers and decimals to the nearest hundredth.
a = 3, b = 5, c = −1.
Discriminant: 25 − 4(3)(−1) = 25 + 12 = 37.
x = −5 ± √376, with √37 ≈ 6.083.
x ≈ 1.0836 ≈ 0.18 or x ≈ −11.0836 ≈ −1.85.
Answer: x = (-5 ± √37)/6, so x ≈ 0.18 or x ≈ -1.85.
Example 3: Time to hit the ground
Test-style
A ball is thrown upward from a 6-foot platform. Its height is h = −16t2 + 48t + 6 feet after t seconds. When does it hit the ground, to the nearest hundredth of a second?
Ground means h = 0: −16t2 + 48t + 6 = 0, so a = −16, b = 48, c = 6.
A quadratic equation is the problem, such as x2 − 3x − 10 = 0. The quadratic formula is one tool for solving it.
Quick check: 3 questions
Question 1 of 3
In 5x2 − x − 4 = 0, what are a, b and c?
Show answer and explanation
Answer: B. Coefficients keep their signs: a = 5, b = −1, c = −4. The first choice dropped the signs, the third reversed the order, and b is the number −1, not the term −x.
Question 2 of 3
What are the solutions of 2x2 + x − 6 = 0?
Show answer and explanation
Answer: C.D = 1 + 48 = 49, so x = −1 ± 74: 64 = 32 or −84 = −2. The first choice has the signs flipped (used +b), the second forgot to divide by 2a, and the last divided by 2 instead of 4.
Question 3 of 3
Using the formula on x2 + 2x + 7 = 0, what happens?
Show answer and explanation
Answer: D.b2 − 4ac = 4 − 28 = −24, and a negative number has no real square root. One real solution needs D = 0, two need D > 0, and division by zero only happens if a = 0.
Quadratic Formula: frequently asked questions
When should I use the quadratic formula?
Use it when a quadratic equation does not factor easily, when the numbers are decimals or large, or when you need exact irrational answers. It works for every quadratic. If the equation factors quickly, or has no x term (such as x2 = 50), factoring or taking square roots is usually faster.
Where does the quadratic formula come from?
It comes from completing the square on ax2 + bx + c = 0. Divide by a, move ca to the other side, add (b2a)2 to both sides, write the left side as (x + b2a)2, take square roots and solve for x. The result is the formula.
How do I remember the quadratic formula?
Say it in one breath: “x equals negative b, plus or minus the square root of b squared minus four a c, all over two a.” Many students sing it to the tune of “Pop Goes the Weasel”. Writing it once at the top of your work, before substituting, also prevents sign errors.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-REI.B.4.a, CCSS HSA-REI.B.4.b
Cite this entry
Effortless Math Team. “Quadratic Formula.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/quadratic-formula/
Flashcard
TermQuadratic Formulasolves ax2 + bx + c = 0
Definitionx = −b ± √b2 − 4ac2a. For x2 + 2x − 8 = 0: −2 ± 62, so 2 or −4.
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