How to Use the Quadratic Formula: Every Case for 2026

The quadratic formula is the single most reliable tool in Algebra 1. Every quadratic equation can be solved with it, no exceptions, no special cases. Factoring is faster when it works, but the quadratic formula always works, and most state tests intentionally use quadratics that do not factor cleanly so students must reach for the formula.

This guide gives you the formula, the discriminant test, four worked examples covering every type of root, and the five mistakes that drag down quiz grades.

The Formula

For any quadratic equation ax² + bx + c = 0 with a ≠ 0:

\[x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\]

Memorize it word for word. The mistake that costs more points than any other is misremembering the formula.

A handy mnemonic: “Negative b plus or minus the square root of b-squared minus four-a-c, all over two-a.”

Step-by-Step Recipe

1. Write the equation in standard form ax² + bx + c = 0. If it is not, move every term to one side.
2. Identify a, b, and c. Sign matters; copy the signs carefully.
3. Plug into the formula. Put each value in parentheses.
4. Compute the discriminant b² − 4ac first.
5. Simplify the square root.
6. Apply ± to get two solutions.
7. Simplify or convert to decimal as the problem requires.

Spend five seconds checking each value of a, b, and c before plugging in. Most errors happen at step 2, not step 5.

The Discriminant: A 5-Second Preview

The expression under the square root, b² − 4ac, is called the discriminant. It tells you the type of roots before you finish computing.

Use the discriminant to predict whether to bother trying to factor first.

Example 1: Two Rational Roots

Solve x² − 7x + 10 = 0.

a = 1, b = −7, c = 10.

Discriminant: (−7)² − 4(1)(10) = 49 − 40 = 9. Perfect square; expect clean roots.

Quadratic formula: x = (7 ± √9) / 2 = (7 ± 3) / 2. So x = 5 or x = 2.

Check by factoring: (x − 5)(x − 2) = 0. Same roots. The discriminant told us this would factor.

Example 2: Two Irrational Roots

Solve x² − 4x + 1 = 0.

a = 1, b = −4, c = 1.

Discriminant: (−4)² − 4(1)(1) = 16 − 4 = 12.

Quadratic formula: x = (4 ± √12) / 2.

Simplify √12 = 2√3: x = (4 ± 2√3) / 2 = 2 ± √3.

Solutions: x = 2 + √3 and x = 2 − √3. Leave in exact form.

Example 3: One Repeated Root

Solve x² − 6x + 9 = 0.

a = 1, b = −6, c = 9.

Discriminant: 36 − 36 = 0.

Quadratic formula: x = (6 ± 0) / 2 = 3.

One repeated root at x = 3. This is the same as (x − 3)² = 0.

Example 4: Complex Roots

Solve x² + 2x + 5 = 0.

a = 1, b = 2, c = 5.

Discriminant: 4 − 20 = −16.

Quadratic formula: x = (−2 ± √(−16)) / 2 = (−2 ± 4i) / 2 = −1 ± 2i.

Solutions: x = −1 + 2i and x = −1 − 2i. Complex roots always come in conjugate pairs for real-coefficient quadratics.

When to Use the Quadratic Formula vs. Other Methods

A practical rule: if you spend more than 30 seconds trying to factor, switch to the quadratic formula.

Watch the Signs

The single most common error is sign confusion when b is negative or c is negative.

For x² − 3x − 4 = 0: – a = 1, b = −3, c = −4. – −b = +3. – b² = 9 (positive whether b was +3 or −3). – −4ac = −4(1)(−4) = +16.

x = (3 ± √(9 + 16)) / 2 = (3 ± 5) / 2 → x = 4 or x = −1.

Plug values in with explicit parentheses to keep signs straight.

Quadratic Equations That Need Rearranging

Sometimes the equation does not start in standard form.

Solve 2x² = 5x + 12.

Move everything to one side: 2x² − 5x − 12 = 0.

Now a = 2, b = −5, c = −12. Discriminant: 25 + 96 = 121 = 11².

x = (5 ± 11) / 4. So x = 4 or x = −3/2.

Always set the equation equal to zero before reading off a, b, c.

Word Problems With the Quadratic Formula

Most quadratic word problems involve area, projectile motion, or revenue. The recipe:

1. Write an equation modeling the situation.
2. Move everything to one side to get standard form.
3. Apply the quadratic formula.
4. Discard any solution that does not make sense (negative time, negative length).

A rectangle has length 3 more than its width. Its area is 40. Find the dimensions.

Let w = width. Length = w + 3. Area: w(w + 3) = 40 → w² + 3w − 40 = 0.

a = 1, b = 3, c = −40. Discriminant: 9 + 160 = 169 = 13².

w = (−3 ± 13) / 2 = 5 or −8.

Width must be positive, so w = 5. Length = 8. Dimensions: 5 by 8.

Common Mistakes

1. Sign errors on b. −b means change the sign of whatever b is. If b = −3, then −b = +3.
2. Forgetting to set the equation equal to zero. You cannot apply the formula to 2x² = 5x + 12 directly.
3. Dropping the ±. Quadratics have two roots most of the time; do not lose one.
4. Mishandling a perfect-square discriminant. √16 = 4, not 8. Take the actual square root.
5. Reporting complex roots wrong. √(−4) = 2i, not 2 ± i.

A Memorable Sequence

Five words to keep on a sticky note:

Identify, plug, discriminant, root, simplify.

Identify a, b, c. Plug into the formula. Compute the discriminant. Take the square root. Simplify or apply ±. Five words. Five points on the rubric.

Frequently Asked Questions

What is the quadratic formula? The quadratic formula is x = (−b ± √(b² − 4ac)) ÷ 2a, and it solves any equation of the form ax² + bx + c = 0 with a ≠ 0. The ± gives up to two solutions. For x² − 5x + 6 = 0, a = 1, b = −5 and c = 6, so x = (5 ± √1) ÷ 2, giving x = 3 or x = 2.

When do you use the quadratic formula? Use the quadratic formula when a quadratic does not factor easily, when the coefficients are large or decimals, or when you need exact irrational or complex answers. Check the discriminant b² − 4ac first: if it is a perfect square, factoring may be faster. For x² − 2x − 1 = 0, the discriminant is 8, so the formula gives x = 1 ± √2.

How do you use the quadratic formula to solve for x? Write the equation in standard form, identify a, b and c, compute the discriminant, then substitute. For 3x² + 5x + 2 = 0: a = 3, b = 5, c = 2, and b² − 4ac = 25 − 24 = 1. So x = (−5 ± 1) ÷ 6, which gives x = −4/6 = −2/3 or x = −6/6 = −1. Put each value in parentheses to protect the signs.

What is the discriminant in the quadratic formula? The discriminant is b² − 4ac, the part under the square root. If it is positive, there are two real roots; if it is zero, there is one repeated root; if it is negative, there are two complex roots. For 4x² − 8x − 60 = 0, it equals 64 + 960 = 1,024, a perfect square (32²), so the roots are rational: x = 5 and x = −3.

What is the quadratic formula derived from? The quadratic formula comes from completing the square on the general equation ax² + bx + c = 0. You divide by a, move c/a to the other side, add (b/2a)² to both sides to make a perfect square, then take the square root and solve for x. The result, x = (−b ± √(b² − 4ac)) ÷ 2a, works for every quadratic.

How do you use the quadratic formula on a calculator? Compute the discriminant first, store or note it, then evaluate each root separately. For 2x² + 3x − 4 = 0, type (−3 + √(3² − 4×2×(−4))) ÷ (2×2) and then the same with a minus sign. That gives about 0.85 and −2.35. Wrap the whole numerator and the whole denominator in parentheses so the calculator does the division correctly.

Closing Thought

The quadratic formula is a five-word recipe and one big rule: set the equation to zero first. Master the sign tracking, glance at the discriminant before you compute, and write the ± every time. Twenty problems of practice and the formula becomes faster than factoring for any messy quadratic.

For more practice, browse our Algebra 1 worksheets and our full Math Topics library. When you are ready for a structured workbook, our Algebra 1 collection drills the quadratic formula and every related topic above.