Quadratic Formula Calculator: Solve Any Quadratic Equation With Steps

The quadratic formula works on every quadratic equation, even ones that will not factor. This guide shows how to find a, b and c, how to read the discriminant, how to simplify messy answers and how to check any result in seconds. Every example below was checked with a computer algebra system.

Note: Calc Solver (calc-solver.co) is an independent guide site and is not affiliated with any calculator mentioned here.

What The Quadratic Formula Is

The quadratic formula is a general solution for any equation of degree 2. It gives the values of x that make ax² + bx + c equal to zero.

x = (−b ± √(b² − 4ac)) / 2a

Here is what each part means:

• a is the number in front of x². It can be any number except 0.
• b is the number in front of x.
• c is the constant, the number with no x.
• ± means you compute the formula twice: once with plus, once with minus. That is why a quadratic has two roots.
• b² − 4ac is the discriminant. It sits under the square root and decides what kind of answer you get.

The answers are called roots, solutions or zeros. On a graph, real roots are the x-intercepts, the points where the parabola crosses the x-axis. Math is Fun's quadratic equation guide has a clear visual of this.

How To Find A, b And c

To find a, b and c, first write the equation in standard form: ax² + bx + c = 0, with everything on one side and 0 on the other. Then read the numbers in order. Most wrong answers from a quadratic formula calculator come from this step, not the formula.

Three rules cover almost every case:

1. Keep the sign with the number. In 2x² − 7x + 3, b is −7, not 7.
2. A missing term means 0. If there is no x term, b = 0. If there is no constant, c = 0.
3. A lone x² means a = 1. The same goes for −x², where a = −1.

Vertex form or factored form? Expand it first. Vertex form is a(x − h)² + k. Factored form is a(x − p)(x − q).

For example, 2(x − 3)² − 8 expands to 2(x² − 6x + 9) − 8, which is 2x² − 12x + 10. So a = 2, b = −12 and c = 10. The roots are x = 1 and x = 5.

Factored form is even quicker, because the roots are already visible. In (x − 2)(x + 5) = 0, the roots are 2 and −5. Expanding gives x² + 3x − 10, which is handy when you need to check a solver's result.

How To Use The Quadratic Formula Calculator

Using the calculator above takes three steps:

1. Pick a tab. Use "a, b, c" if you already know the coefficients. Use "Type equation" if your equation is not in standard form. Use "Vertex or factored form" for a(x − h)² + k or a(x − p)(x − q).
2. Enter your numbers. Negatives need the minus sign. Fractions like 1/3 and decimals like 0.25 both work.
3. Read the result. You get the discriminant, the roots in exact form and as decimals, the vertex and a graph.

The steps show how the formula was applied, so you can copy the method onto homework. If a is 0, the tool tells you the equation is linear. For that case, the equation solver handles linear equations, systems and inequalities.

Solving With The Quadratic Formula Step By Step

Solving by hand follows the same order every time: find a, b and c, work out the discriminant, then finish the formula with plus and with minus. Here is 2x² − 7x + 3 = 0.

Step 1: Name the coefficients. a = 2, b = −7, c = 3.

Step 2: Find the discriminant. b² − 4ac = (−7)² − 4(2)(3) = 49 − 24 = 25.

Step 3: Take the square root. √25 = 5.

Step 4: Fill in the formula. x = (−(−7) ± 5) / (2 × 2) = (7 ± 5) / 4.

Step 5: Split into two answers. x = 12/4 = 3, and x = 2/4 = 1/2.

Step 6: Check. 2(3)² − 7(3) + 3 = 18 − 21 + 3 = 0. It works.

The most common slip is Step 4. The formula starts with −b, so when b is negative, −b is positive. Wrap negative numbers in parentheses when squaring too: (−7)² is 49, but typing −7² into many calculators gives −49.

What The Discriminant Tells You

The discriminant, b² − 4ac, tells you how many roots there are and what type they are before you finish the problem. Check it first and you will know what kind of answer to expect.

A perfect-square discriminant is a useful signal. It means the equation could have been factored, so you can double-check with factoring if you like.

Solving Quadratics With Complex Roots

When the discriminant is negative, the roots are complex numbers written as p ± qi, where i = √(−1). The quadratic still has two answers. They just are not on the real number line.

Take x² + 2x + 5 = 0:

1. a = 1, b = 2, c = 5.
2. D = 2² − 4(1)(5) = 4 − 20 = −16.
3. √(−16) = √16 × √(−1) = 4i.
4. x = (−2 ± 4i) / 2 = −1 ± 2i.

Complex roots always come in pairs, called conjugates: −1 + 2i and −1 − 2i. When a, b and c are real, you will never get just one complex root.

If your class has not covered imaginary numbers yet, the expected answer is usually "no real solution." Read the question carefully.

Simplifying Your Answer

A quadratic formula calculator often gives a messy exact answer, and teachers usually want it simplified. Simplifying means pulling perfect squares out of the radical, then reducing the fraction.

Take 2x² + 4x − 3 = 0. The discriminant is 16 + 24 = 40, so:

x = (−4 ± √40) / 4

1. Simplify the radical. √40 = √(4 × 10) = 2√10.
2. Rewrite. x = (−4 ± 2√10) / 4.
3. Divide every term by the common factor 2. x = (−2 ± √10) / 2.

As decimals, the roots are about 0.581 and −2.581.

One rule trips people up: you can only cancel a factor if it divides every term on top. In (−4 ± 2√10) / 4, you cannot cancel the 4 with just the −4. If you need help reducing the final fraction, the fraction calculator shows each step.

Finding The Vertex, Axis Of Symmetry And Graph

The vertex is the highest or lowest point of the parabola. Its x-value is x = −b / 2a, which is also the midpoint between the two roots. Plug that x back in to get the y-value.

For y = 2x² − 7x + 3:

• Axis of symmetry: x = −(−7) / (2 × 2) = 7/4 = 1.75
• Vertex: y = 2(1.75)² − 7(1.75) + 3 = −3.125, so the vertex is (1.75, −3.125)
• Y-intercept: c = 3, so the graph crosses the y-axis at (0, 3)
• Direction: a = 2 is positive, so the parabola opens upward and the vertex is a minimum

Notice that the roots, 1/2 and 3, sit exactly 1.25 units on each side of 1.75. That symmetry is a fast way to spot a wrong root.

Where The Quadratic Formula Comes From

The quadratic formula comes from solving ax² + bx + c = 0 by completing the square. Knowing this explains why the formula works on every quadratic, not just ones that factor.

1. Divide by a: x² + (b/a)x + c/a = 0
2. Move c/a: x² + (b/a)x = −c/a
3. Add (b/2a)² to both sides: x² + (b/a)x + (b/2a)² = (b/2a)² − c/a
4. Write the left side as a square: (x + b/2a)² = (b² − 4ac) / 4a²
5. Take the square root of both sides: x + b/2a = ±√(b² − 4ac) / 2a
6. Subtract b/2a: x = (−b ± √(b² − 4ac)) / 2a

Step 5 is where the ± comes from. Every positive number has two square roots, so there are two answers.

Quadratic Formula Vs Other Methods

The quadratic formula always works, but it is not always the fastest. Pick the method that fits the equation in front of you.

A good habit: check the discriminant first. If it is a perfect square, try factoring. If not, go straight to the formula.

Solving Quadratics On A TI-84 Or Casio Calculator

Most graphing and scientific calculators have a built-in polynomial solver, so you only need to enter a, b and c.

TI-84 Plus CE: press the apps key, choose PlySmlt2, then Polynomial Root Finder. Set the order to 2, enter a, b and c, and select Solve. The app can switch the roots between fractions and decimals, as shown in Texas Instruments' Polynomial Root Finder example.

Casio fx-991EX ClassWiz: press MENU, pick Equation/Func, choose Polynomial, then degree 2. Type each coefficient and press equals after it, then press equals again to solve. Use the arrow keys to move between x1 and x2. The Casio ClassWiz quick start guide covers the same menu.

Casio fx-991ES PLUS: press MODE, 5 (EQN), then 3 for aX² + bX + c = 0. Enter the coefficients and press equals.

If your calculator has no solver, type the formula with parentheses: (−b + √(b² − 4ac)) ÷ (2a). The parentheses around 2a matter. Without them, the calculator divides by 2 and then multiplies by a. The scientific calculator on this site handles this format too.

Real-World Problems The Quadratic Formula Solves

Quadratics show up whenever something speeds up, slows down or depends on a square. Here are three common setups.

A thrown ball. A ball is thrown up from 5 feet at 64 feet per second. Its height is h = −16t² + 64t + 5. To find when it lands, solve −16t² + 64t + 5 = 0. The roots are about 4.077 and −0.077. Time cannot be negative, so the ball lands after about 4.08 seconds.

An area problem. A rectangle is 3 feet longer than it is wide and has an area of 40 square feet. That gives w(w + 3) = 40, or w² + 3w − 40 = 0. The roots are 5 and −8. A width cannot be negative, so the rectangle is 5 feet by 8 feet.

The golden ratio. The golden ratio is the positive root of x² − x − 1 = 0. The formula gives x = (1 + √5) / 2 ≈ 1.618.

In word problems, always ask whether both roots make sense. Lengths, times and counts usually rule one out.

How To Check Your Roots

You can check any answer from a quadratic formula calculator in two ways, and the second one takes seconds.

Method 1: Substitute. Plug each root into the original equation. Both sides should be equal, or very close if you rounded.

Method 2: Use the sum and product of roots. For ax² + bx + c = 0, the two roots always add up to −b/a and multiply to c/a. For 2x² − 7x + 3 = 0, the roots 3 and 1/2 add to 3.5, which is 7/2, and multiply to 1.5, which is 3/2. Both match, so the answer is right.

The sum-and-product test also works on irrational and complex roots, where substituting gets messy.

When A Calculator Gives A Wrong Answer

Even a working calculator can lose accuracy when b is huge compared with a and c. The formula subtracts two numbers that are almost the same, and the small root loses most of its digits.

Take x² + 100,000,000x + 1 = 0. The true small root is about −0.00000001. Computing the formula directly in standard double-precision numbers gives about −0.0000000075, which is 25% off.

The fix is a second version of the formula for the small root: x = 2c / (−b − √(b² − 4ac)). It avoids the subtraction and gives −0.00000001. The calculator above uses this stable method automatically. If you code your own solver for a class project, use it too.

Common Quadratic Formula Mistakes

Most mistakes come from signs and order of operations, not from the formula itself. This table shows the fix for each one.

Summary

• The quadratic formula, x = (−b ± √(b² − 4ac)) / 2a, solves every equation in the form ax² + bx + c = 0 with a ≠ 0.
• Write the equation in standard form first, then read a, b and c with their signs.
• The discriminant tells you the answer type: positive means two real roots, zero means one, negative means two complex roots.
• Simplify radicals and divide every term by the common factor for a clean exact answer.
• The vertex sits at x = −b/2a, halfway between the roots.
• Check any answer with the sum (−b/a) and product (c/a) of the roots.

Next step: enter your own a, b and c in the quadratic formula calculator at the top of this page. For other equation types, try the math solver, or browse all tools on the Calc Solver homepage.

Frequently Asked Questions