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:

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

The Ultimate Algebra Bundle 2026: From Pre-Algebra to Algebra II

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

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.

The Ultimate Middle School Math Bundle: Grades 6-8

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

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.

The Ultimate Elementary School Math Bundle: Grade 3-5

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

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

Do I have to memorize the quadratic formula?
Yes. It appears on Algebra 1, Algebra 2, pre-calc, SAT, ACT, and many AP exams. Memorize it word for word.

Is the quadratic formula on the SAT reference sheet?
No. The Digital SAT does not provide a reference sheet of formulas like the SAT did in older formats. You need this one in memory.

How is the quadratic formula derived?
By completing the square on the general quadratic ax² + bx + c = 0. See our completing the square guide for the full derivation.

What is the fastest way to spot complex roots?
Compute the discriminant. If it is negative, the roots are complex.

Can the quadratic formula give a single answer?
Yes, when the discriminant is zero. That gives one repeated root.

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.

Pre-Algebra for Beginners 2026: The Ultimate Step by Step Guide to Preparing for the Pre-Algebra Test

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.