How to Use the Quadratic Formula: A Beginner’s Walk-Through

The quadratic formula is the one tool that solves any quadratic equation — even the ones that won’t factor. Memorize it, learn to use it cleanly, and you can handle quadratics on the SAT, ACT, GED, and beyond.

The formula

For any quadratic in the form $ax^2 + bx + c = 0$:

$$x = \dfrac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$

The $\pm$ gives you two answers (the two roots of the parabola).

The 4-step routine

1. Get the equation in standard form ($ax^2 + bx + c = 0$).
2. Identify $a$, $b$, and $c$.
3. Plug into the formula — carefully with signs.
4. Simplify the square root, then split into two answers.

Worked example

Solve $2x^2 – 5x – 3 = 0$.

1. Standard form: ✓.
2. $a = 2$, $b = -5$, $c = -3$.
3. Plug in: $x = \dfrac{-(-5) \pm \sqrt{(-5)^2 – 4(2)(-3)}}{2(2)} = \dfrac{5 \pm \sqrt{25 + 24}}{4} = \dfrac{5 \pm \sqrt{49}}{4} = \dfrac{5 \pm 7}{4}$.
4. Two answers: $x = \tfrac{12}{4} = 3$ or $x = \tfrac{-2}{4} = -\tfrac{1}{2}$.

The discriminant — peek before you solve

The expression under the square root, $b^2 – 4ac$, is called the discriminant. It tells you what kind of roots you’ll get:

• Positive → two distinct real solutions.
• Zero → one repeated real solution.
• Negative → two complex solutions (no real roots).

Use it to know in advance whether your answer will be clean.

When to use the formula vs. factoring

Common mistakes

• Forgetting the negative on $-b$.
• Squaring $b$ as $-b^2$ instead of $b^2$. ($(-5)^2 = 25$, not $-25$.)
• Forgetting the $\pm$ — and missing half your answer.

FAQ

What is the quadratic formula?

$x = \dfrac{-b \pm \sqrt{b^2 – 4ac}}{2a}$, which solves any equation in the form $ax^2 + bx + c = 0$.

When should I use the quadratic formula instead of factoring?

When the quadratic doesn’t factor cleanly, or when factoring would take longer than just plugging in.

What does the discriminant tell me?

It tells you whether you’ll get two real roots, one repeated root, or two complex roots.

Is the quadratic formula on the SAT?

Yes, but the SAT prefers values that do factor — so always check for an easy factoring first.

Can I use a calculator for the quadratic formula?

Yes — and many graphing calculators have a built-in solver.

How does completing the square relate to the quadratic formula?

The quadratic formula is what you get when you complete the square on the general form $ax^2 + bx + c = 0$. Every quadratic-formula calculation is really a completed square in disguise.

How do I know when my answer involves complex numbers?

When the discriminant $b^2 – 4ac$ is negative. The square root of a negative number isn’t real — it’s a complex number written with $i = \sqrt{-1}$. For SAT/ACT purposes you’ll rarely see complex roots; for Algebra 2 and beyond, they appear regularly.

What’s the vertex of a parabola, and how do I find it?

The vertex is the highest or lowest point of the parabola. Its x-coordinate is $x = -\tfrac{b}{2a}$. Plug that back into the original equation to find the y-coordinate. The vertex tells you the maximum (if $a < 0$) or minimum (if $a > 0$) value the quadratic takes.

How does the sign of $a$ change the parabola?

$a > 0$ → parabola opens up (smile-shaped). $a < 0$ → parabola opens down (frown-shaped). The bigger $|a|$ is, the narrower the parabola.

Worked example with a negative discriminant

Solve $x^2 + 2x + 5 = 0$.

• $a = 1$, $b = 2$, $c = 5$.
• Discriminant: $4 – 20 = -16$.
• $\sqrt{-16} = 4i$.
• $x = \dfrac{-2 \pm 4i}{2} = -1 \pm 2i$.

Two complex roots: $-1 + 2i$ and $-1 – 2i$. The parabola never crosses the x-axis.

Worked example with a repeated root

Solve $x^2 – 6x + 9 = 0$.

• $a = 1$, $b = -6$, $c = 9$.
• Discriminant: $36 – 36 = 0$.
• $x = \dfrac{6 \pm 0}{2} = 3$.

One repeated root: $x = 3$. The parabola just touches the x-axis at a single point. Always double-check by noticing the perfect-square trinomial: $x^2 – 6x + 9 = (x – 3)^2$.

Where quadratics appear in the real world

• Projectile motion. A ball thrown straight up follows the equation $h = -16t^2 + v_0 t + h_0$. Solving for when it lands ($h = 0$) is a quadratic-formula problem.
• Profit maximization. Revenue × quantity sold often produces a quadratic; the vertex tells you the price that maximizes profit.
• Geometry. Diagonal lengths, areas, and Pythagorean problems often collapse into quadratics.
• Physics. Energy, motion, and oscillation formulas frequently involve $x^2$.

In every case, the quadratic formula is your universal tool.

Test-day cheat sheet

• Always check first if the quadratic factors easily.
• If factoring fails, write down $a$, $b$, $c$ carefully (watch the signs).
• Compute the discriminant first — it tells you what kind of answer to expect.
• Don’t forget the $\pm$. Two answers, almost always.

Vieta’s formulas — a hidden shortcut

For a quadratic $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$:

• Sum of roots: $r_1 + r_2 = -\dfrac{b}{a}$
• Product of roots: $r_1 \cdot r_2 = \dfrac{c}{a}$

This lets you check your answers fast. For $x^2 – 5x + 6 = 0$, the roots should sum to 5 and multiply to 6. The roots are 2 and 3. ✓

Vieta’s also lets you build a quadratic given its roots: if the roots are 4 and $-3$, then $b = -(4 + (-3))(1) = -1$ and $c = 4 \cdot (-3) = -12$, so the quadratic is $x^2 – x – 12 = 0$.

Comparing all 4 methods

Which method should you use? Here’s a decision tree:

1. Can you factor it in 10 seconds? Use factoring.
2. Is $b$ even and $a = 1$? Try completing the square — it’ll be clean.
3. Is the equation already $(\text{something})^2 = \text{number}$? Take square roots directly.
4. Otherwise: use the quadratic formula. It always works.

The quadratic formula is the universal hammer. The other methods are faster when they apply.

Practice set

Solve each:

1. $x^2 – 7x + 12 = 0$
2. $2x^2 + 5x – 3 = 0$
3. $x^2 + 4x + 5 = 0$
4. $3x^2 – 6x + 2 = 0$
5. $x^2 – 9 = 0$
6. $4x^2 = 12x$

Answers: 1) $x = 3, 4$. 2) $x = \tfrac{1}{2}, -3$. 3) $x = -2 \pm i$. 4) $x = 1 \pm \tfrac{\sqrt{3}}{3}$. 5) $x = \pm 3$. 6) $x = 0, 3$.

Final note

The quadratic formula isn’t scary — it’s just a very compressed recipe. Write it down at the start of every test. Plug in the numbers carefully. Double-check signs. You’ll be right far more often than the students who try to memorize tricks.

Extra study tips that move the needle

Most students don’t fail because the math is too hard — they fail because their practice habits are inefficient. Here are the habits that separate the students who improve fast from those who stall.

Practice with a timer. Untimed practice teaches you to eventually get the right answer; timed practice teaches you to get it in test conditions. Set a stopwatch every time you sit down. Aim for 90 seconds per question on most standardized tests.

Keep an error log. A simple spreadsheet with three columns — Problem, My answer, Correct answer, Why I missed it — is the single most powerful study tool ever invented. Review your error log weekly. The same mistakes show up again and again until you name them.

Mix topics every session. Doing 20 problems on the same topic feels productive, but spaced and interleaved practice — mixing topics — builds retrieval skills, which is what the test actually measures. Spend 70% of your time on mixed sets and only 30% on isolated drills.

Sleep on it. Memory consolidation happens during sleep. A 30-minute session the night before a quiz, followed by 7+ hours of sleep, beats a 3-hour cram session that ends at midnight. This is settled cognitive science.

Teach the topic out loud. If you can’t explain it, you don’t fully know it. Either record yourself, write a one-paragraph “how I’d teach this” explanation, or grab a friend to listen. Teaching exposes the gaps your problem sets hid.

When to ask for help

Spinning your wheels for more than 15 minutes on a single problem is a signal — not of failure, but of a missing piece of background. Stop, mark the problem, and either ask a teacher, post in our community, or watch a video on the relevant subtopic. Resuming after gaining the missing piece is much more efficient than guessing your way forward.

A quick self-assessment

Before you close this tab, answer these three questions honestly:

1. What’s the one topic in this article you understood best?
2. What’s the one topic that still feels fuzzy?
3. What concrete next step (a worksheet, a practice test, a video) will you take in the next 48 hours?

Writing those answers down — even just in a notes app — has been shown to roughly double the chance you actually follow through. Treat the next 48 hours as a small, doable experiment, not a marathon. Your future test-day self will thank you.

Drill it with our Algebra 1 worksheets or the complete Algebra Bundle.