Quadratic Equation Solver (Free Step-by-Step Tool)

Use this free quadratic equation solver to solve any equation in the form ax² + bx + c = 0. Enter the three coefficients and instantly get the roots (real or complex), the discriminant, the vertex and axis of symmetry, the factored form, a live parabola graph, and a clear step-by-step solution.

How to use the quadratic equation solver

1. Write your equation in standard form, ax² + bx + c = 0, so the right-hand side is zero.
2. Enter a, b, and c in the three boxes. Decimals and negatives are fine; a cannot be 0 (that would make the equation linear).
3. Press Solve. You will see the roots, discriminant, vertex, axis of symmetry, y-intercept, factored form, a graph, and the full working.
4. Share or save: the page link updates with your coefficients (for example ?a=2&b=5&c=-3), and the Copy results button copies everything to your clipboard.

Worked examples

Two real roots: For 2x² + 5x − 3 = 0, the discriminant is 5² − 4(2)(−3) = 49, so x = 1/2 or x = −3.

One repeated (double) root: For x² − 4x + 4 = 0, the discriminant is 0, so x = 2 is a double root and the parabola just touches the x-axis.

Complex roots: For x² + x + 1 = 0, the discriminant is −3, so there are no real roots; instead x = −0.5 ± (√3/2)i.

What the solver shows you

• Roots: the solutions of the equation, shown as exact fractions when possible.
• Discriminant (b² − 4ac): tells you whether the roots are two real, one repeated, or two complex.
• Vertex and axis of symmetry: the turning point of the parabola and its line of symmetry.
• Y-intercept and factored form: where the curve meets the y-axis, and the equation rewritten as a product of factors.
• Graph: the parabola with roots and vertex marked.

Frequently asked questions

What is the solution of a quadratic equation?

The solutions, or roots, of a quadratic equation are the x-values that make ax² + bx + c equal zero. A quadratic can have two real solutions, one repeated solution, or two complex solutions, depending on the discriminant b² − 4ac. On a graph, real solutions are the points where the parabola crosses the x-axis. For example, x² − 9 = 0 has the solutions x = 3 and x = −3.

How do you solve a quadratic equation for x?

First rewrite the equation in standard form, ax² + bx + c = 0. Then factor, complete the square, or use the quadratic formula. For 2x² + 5x − 3 = 0, the discriminant is 5² − 4(2)(−3) = 49, so x = (−5 ± 7)/4, which gives x = 1/2 or x = −3. Plug each answer back into the original equation to check it.

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 expression under the square root, b² − 4ac, is the discriminant. If it is positive there are two real roots, if it is zero there is one double root, and if it is negative the two roots are complex.

How do you solve a quadratic equation by factoring?

Write the quadratic as a product of two binomials and set each factor equal to zero. For x² − 5x + 6 = 0, look for two numbers that multiply to 6 and add to −5: −2 and −3. So (x − 2)(x − 3) = 0, which gives x = 2 or x = 3. Not every quadratic factors neatly with whole numbers, so the quadratic formula is the backup that always works.

How do you solve a quadratic equation on a calculator?

Put the equation in standard form, identify a, b and c, and enter them into the solver, keeping the signs. For x² − 4x + 4 = 0, enter a = 1, b = −4 and c = 4. The solver returns the double root x = 2, a discriminant of 0, the vertex and a graph. On a handheld calculator, you can type the quadratic formula directly, using parentheses around b² − 4ac and around 2a.

How can you tell if an equation is quadratic?

An equation is quadratic if, after simplifying, the highest power of the variable is 2 and that term’s coefficient is not zero. So 3x² − x + 7 = 0 is quadratic, while 4x + 1 = 0 is linear and x³ − 2x = 0 is cubic. Watch for hidden squares: x(x + 2) = 5 expands to x² + 2x − 5 = 0, which is a quadratic equation.

Want the full lesson with methods, worked examples, and a free worksheet? Read How to Solve a Quadratic Equation.

How to use the Quadratic Equation Solver for homework

The Quadratic Equation Solver is most useful when you treat it as a learning check, not just a shortcut to the final solution. Start by copying the original problem carefully, including signs, exponents, decimal points, fractions, parentheses, and units. Then enter the values in the same order the problem gives them. A small typing change can completely change the result, especially in algebra, statistics, geometry, and probability problems.

Before you press the button to solve, make a quick estimate or prediction. The estimate does not need to be exact. Its job is to help you notice impossible answers. If a distance becomes negative, a probability is bigger than 1, an angle looks too large, or a decimal point seems misplaced, go back and check the input before trusting the final result.

Before you enter the problem

• Rewrite the problem in a clean line so every value is easy to see.
• Use parentheses around grouped expressions, especially in fractions and exponents.
• Keep units with the numbers while you work, even if the solver only asks for the numbers.
• Check whether the problem wants an exact value, a decimal approximation, or a rounded answer.
• Look for restrictions such as positive values only, a chosen interval, or a required domain.

How to read the result

After the solver gives a result, read more than the final line. If steps, tables, graphs, or intermediate values are shown, use them to understand how the answer was built. That is especially important when you are studying for a quiz or test, because teachers often give more credit for a correct process than for an unsupported number.

Try to identify the main idea behind the result. For example, ask yourself which formula was used, which operation changed the expression, which value controlled the graph, or which assumption made the answer possible. When you can explain that idea in your own words, the tool has helped you learn the skill instead of only checking one problem.

Common mistakes to avoid

Turn the answer into practice

One good way to study is to solve the problem by hand first, then use the Quadratic Equation Solver to check your work. If your answer is wrong, do not erase everything immediately. Find the first line where your work stops matching the calculator’s logic. That line is usually where the real misunderstanding happened.

1. Work the problem on paper and circle your final answer.
2. Use the solver to check the result.
3. If the answers match, write one sentence explaining the method.
4. If they do not match, compare each step until you find the first difference.
5. Redo a similar problem without the tool to make sure the correction sticks.

When to use a calculator and when to work by hand

Use the Quadratic Equation Solver when you want to check a long calculation, explore a pattern, test a graph, or confirm a result after practicing. Work by hand when the assignment asks for steps, when you are learning a new method, or when the test will not allow a digital tool. The strongest students use both: hand work to build understanding and calculators to check accuracy.

If you are preparing for a timed test, practice some problems without the tool and some with it. That balance helps you build speed while still understanding the math. Over time, the goal is to need the calculator less often for routine steps and use it more strategically for checking, exploring, and confirming your reasoning.

Quick accuracy checklist

Before you leave the page, use the Quadratic Equation Solver result to answer three quick questions: What was the input? What rule, formula, or method connects the input to the result? What would change if one number in the problem changed? These questions turn a single calculator answer into a reusable math skill.

For homework, write the original problem, the calculator’s result, and one short explanation of the method in your notebook. For test review, cover the result and try to reproduce the setup on your own. If you can set up a similar problem without looking back at the solver, you are much closer to owning the skill.