Introduction
A quadratic equation looks like ax² + bx + c = 0. This quadratic equation solver finds the answers, called roots. Type in the numbers a, b, and c, or type the whole equation. Then press Calculate.
You get more than just an answer. The calculator shows each step, so you can see how the work is done. Pick one of three ways to solve: the quadratic formula, factoring, or completing the square. Switch between them any time to compare.
The tool also gives you:
- The discriminant (Δ = b² − 4ac) and what it tells you about the roots
- Exact answers with square roots, plus decimal answers
- A graph of the parabola with the vertex and intercepts marked
- A number line that shows where the real roots sit
- A summary table with vertex form, factored form, axis of symmetry, domain, and range
Fractions, decimals, and complex (imaginary) roots all work. There is also a "Verify Your Answer" box. Type in the answer you got on paper, and the calculator will tell you if it is right. It is a fast way to check homework and learn algebra at the same time.
How to use our Quadratic Calculator
Enter the numbers from your quadratic equation, and the calculator gives you the roots, the discriminant, the vertex, a graph, and step-by-step work.
Input mode: Pick Coefficient Mode to type a, b, and c in separate boxes. Pick Equation Mode to type the whole equation on one line.
a (x² coefficient): Type the number in front of x². It cannot be 0. You can use whole numbers, decimals, or fractions like 3/4.
b (x coefficient): Type the number in front of x. Use 0 if your equation has no x term. Tap the ± button to flip the sign.
c (constant): Type the plain number at the end. Use 0 if there is no constant term.
Quadratic equation box: In Equation Mode, type your equation, like x^2 - 5x + 6 = 0. Use ^ for powers. If you leave off "= 0", we add it for you.
Show Keyboard: Tap this to get on-screen math keys for x, ^, √, and numbers.
Try an example: Tap any example equation to load and solve it right away.
Quick actions: Tap Solve, Find Vertex, Find Intercepts, Factor, or Complete the Square to jump straight to that answer.
Solve by: Choose Quadratic Formula, Factoring Method, or Completing the Square. The steps change as soon as you pick.
Calculate: Press this button to solve. Press Clear to start over with the default equation.
Result tabs: Use Steps, Graph, Number Line, and Summary to see the work, the parabola, the roots on a line, and all key values.
Verify Your Answer: Type your own roots, like x = 1, x = 2, and press Check Answer to see if you got it right.
What Is a Quadratic Equation?
A quadratic equation is a math sentence where the highest power of the variable is 2. Its standard form is ax² + bx + c = 0, where a, b, and c are numbers and a is not zero. If a were zero, the x² term would vanish and you would have a straight-line (linear) equation instead.
The three numbers are called coefficients. a sits in front of x², b sits in front of x, and c is the constant term. Solving the equation means finding the values of x that make both sides equal. Those values are called the roots or solutions.
The Parabola
When you graph a quadratic, you get a U-shaped curve called a parabola. If a is positive, the curve opens up and the lowest point is the smallest value. If a is negative, the curve opens down and the highest point is the largest value. That turning point is the vertex, found at x = −b ÷ 2a. A vertical line through the vertex is the axis of symmetry, and both halves of the parabola mirror each other across it. The roots are the spots where the curve crosses the x-axis, and the point (0, c) is the y-intercept.
The Discriminant Tells You What to Expect
Before solving, you can check the discriminant: Δ = b² − 4ac. It tells you how many roots the equation has and what kind they are.
- Δ > 0: two different real roots. The parabola crosses the x-axis twice.
- Δ = 0: one repeated real root. The parabola just touches the x-axis.
- Δ < 0: two complex roots that use the imaginary number i. The parabola never touches the x-axis.
Three Ways to Solve
1. Factoring. Break the quadratic into two smaller pieces, like (x − 1)(x − 2) = 0. Then set each piece equal to zero and solve. This is the fastest way, but it only works when the roots are nice fractions or whole numbers.
2. Completing the square. Rewrite the equation so one side is a perfect square, such as (x − h)² = k. Then take the square root of both sides. This method also turns the equation into vertex form, a(x − h)² + k, where (h, k) is the vertex.
3. The quadratic formula. This one always works, no matter how messy the numbers are:
x = ( −b ± √(b² − 4ac) ) ÷ 2a
Plug in a, b, and c, then use the plus sign for one root and the minus sign for the other.
Handy Shortcuts
Vieta's rules let you check your answers fast. The two roots add up to −b ÷ a, and they multiply to c ÷ a. If your roots do not follow both rules, something went wrong.
Every quadratic has a domain of all real numbers, because you can put any number in for x. The range depends on the vertex: it starts at k and goes up when the curve opens up, or stops at k and goes down when the curve opens down.
Where Quadratics Show Up
Quadratic equations model many real things. They describe the path of a thrown ball, the shape of a satellite dish, the arc of a bridge cable, how long an object falls, profit and cost curves in business, and area problems where length and width are linked. Anytime a value grows and then shrinks (or shrinks and then grows), a quadratic is often the right tool.