Quadratic Equation Calculator
Solve quadratic equations by factoring, using the quadratic formula, or completing the square, with clear step-by-step solutions.
How to Use the Quadratic Equation Calculator
This quadratic solver is one of several tools inside our AI Math Solver, which also handles algebra, calculus, geometry, and statistics problems.
Use this quadratic equation calculator to solve equations in the form ax² + bx + c = 0. Enter your equation to find the roots, see the quadratic formula applied step by step, or graph the resulting parabola.
1. Add Your Problem:
Type the equation with the Calculator keypad, or switch to Math Input for proper notation like \(x^2\) and \(\pm\). Canvas lets you handwrite the equation or upload a photo if you're working from paper.
2. Pick a Tool:
The default gives a full step-by-step answer, acting as a roots of quadratic equation calculator that lists both solutions clearly. Choose Create Graph to see a quadratic graph of the parabola — this also works as a quadratic function calculator, showing exactly where the roots sit on the curve. If you've already solved it yourself, Find My Mistake checks your working and points out where it went wrong.
3. Set Your Language:
Open the language dropdown labeled "Default" and pick the language you want your answer in.
4. Get Your Answer:
Click Solve to submit the equation. Your steps, graph, or mistake check appears below, depending on what you chose.
How to Solve a Quadratic Equation
A quadratic equation can be written in standard form as \(ax^2 + bx + c = 0\), where a≠0. Learning how to solve a quadratic equation comes down to picking one of three methods, and the numbers in front of you usually decide which one fits.
Put the Equation in Standard Form
Before solving, rewrite the equation as \(ax^2 + bx + c = 0\). This makes it easier to identify the three coefficients used in the quadratic formula.
1. Factoring
Factoring works when the equation splits neatly into two binomials. You rewrite \(ax^2 + bx + c = 0\) as \((px + q)(rx + s) = 0\), and since two things multiply to zero, at least one of them must be zero.
Learning how to factor a quadratic equation is mostly about spotting a number pair that multiplies to \(ac\) and adds to \(b\). Once you find that pair, how to solve a quadratic equation by factoring takes just a couple of lines. If no such pair exists, move on to another method — forcing it wastes time.
2. Completing the Square
This method rewrites one side of the equation as a perfect square. You turn \(ax^2 + bx + c = 0\) into something like \(a(x + d)^2 = e\), then take the square root of both sides.
It's slower than factoring, but it's the reasoning behind the formula itself, and it's worth knowing before you learn how to graph a quadratic function in vertex form.
3. Using the Quadratic Formula
The quadratic formula solver approach works on every quadratic equation, no exceptions. The quadratic equation solver formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Take \(b\), flip its sign, then add and subtract the square root of the discriminant \(b^2 - 4ac\), and divide the whole thing by \(2a\). To solve for x quadratic problems this way, you don't need clever factoring — just three numbers and careful arithmetic. This is how solving quadratic equations stays reliable even when the roots come out messy or complex.
Understanding the Discriminant
The discriminant is the part of the quadratic formula under the square root:
\[ D = b^2 - 4ac \]
For a quadratic equation in standard form, \(ax^2 + bx + c = 0\), the discriminant tells you what type of roots to expect before you finish solving the equation.
What Does the Discriminant Tell You?
| Discriminant | Type of Roots | What it means |
|---|
| \(b^2 - 4ac > 0\) | Two distinct real roots | The parabola crosses the x-axis twice. |
| \(b^2 - 4ac = 0\) | One repeated real root | The parabola touches the x-axis once. |
| \(b^2 - 4ac < 0\) | Two complex roots | The parabola does not cross the x-axis. |
A positive discriminant gives two different real solutions. A zero discriminant means both solutions are the same, so the equation has one repeated real root. A negative discriminant means there are no real solutions; instead, the equation has two complex roots.
The discriminant is especially useful when using the quadratic formula because it tells you what kind of answer to expect before evaluating the square root.
Worked Example
Solve \(3x^2 + 11x - 4 = 0\) using the quadratic formula.
Step 1 — Identify the coefficients.
\[ a = 3, \quad b = 11, \quad c = -4 \]
These come straight from matching the equation to standard form.
Step 2 — Find the discriminant.
\[ b^2 - 4ac = 11^2 - 4(3)(-4) = 121 + 48 = 169 \]
Because the discriminant is (169), which is positive, the equation has two distinct real roots.
Step 3 — Substitute into the formula.
\[ x = \frac{-11 \pm \sqrt{169}}{2(3)} \]
Every value from Step 1 goes into its place.
Step 4 — Simplify the square root.
\[ x = \frac{-11 \pm 13}{6} \]
Since \(169\) is a perfect square, the root simplifies cleanly to \(13\).
Step 5 — Solve both cases.
\[ x = \frac{-11 + 13}{6} = \frac{1}{3} \qquad x = \frac{-11 - 13}{6} = -4 \]
The \(\pm\) sign always gives two separate answers.
Factoring reaches the same result through \((3x - 1)(x + 4) = 0\), and completing the square lands on the same two values too — only the path changes.
Common Mistakes
Not setting the equation to zero first. The formula assumes \(ax^2 + bx + c = 0\). If your equation is \(3x^2 + 11x = 4\), you have to subtract 4 from both sides before reading off \(c = -4\). Skip that step and you'll use \(c = 4\) by mistake, which throws off the entire discriminant.
Dropping the sign on \(b\). The formula starts with \(-b\), so the sign always flips. In \(3x^2 - 7x + 2 = 0\), \(b = -7\), so \(-b\) becomes \(7\) — not \(-7\). Getting this backward flips the sign of both final roots.
Simplifying the fraction incorrectly. After reaching something like \(\frac{-11 \pm 13}{6}\), don't divide just one term because it "looks even." Reduce the whole numerator and denominator together, or leave the fraction as it is.
If your equation isn't quadratic — for example it involves fractions, exponents, or multiple variables — try the full AI math solver for step-by-step help with any algebra problem.
Questions
Questions Students Ask About Quadratic Equations
What is the quadratic formula?
+The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It solves any equation written as \(ax^2 + bx + c = 0\). You substitute the three coefficients and the formula returns both roots.
How do I solve a quadratic equation?
+Rewrite it as \(ax^2 + bx + c = 0\), then choose factoring, completing the square, or the quadratic formula. Factoring is fastest when the numbers split cleanly, while the formula always works, even when they don't. Checking the discriminant first tells you what kind of roots to expect.
What are the roots of a quadratic equation?
+The roots are the \(x\)-values that make the equation true — the points where the parabola crosses the \(x\)-axis. A quadratic can have two real roots, one repeated root, or two complex roots. You find them by factoring the expression or applying the formula directly.
What does the discriminant tell me?
+The discriminant is \(b^2 - 4ac\), the value under the square root in the formula. A positive discriminant gives two real roots, zero gives one repeated root, and a negative discriminant gives two complex roots. Checking it first saves you from trying to factor something that has no real solution.
Can a quadratic equation have no real solutions?
+Yes, this happens whenever \(b^2 - 4ac\) is negative. The parabola never touches the \(x\)-axis in that case, so there's no real number where the equation equals zero. The two roots still exist, but they involve the imaginary unit \(i\).
How do I factor a quadratic equation?
+Find two numbers that multiply to \(ac\) and add to \(b\), then use them to split the middle term into two parts. Group the four terms into two binomials and set each one equal to zero. This only works when a matching number pair actually exists.
How do I graph a quadratic function?
+Choose several \(x\)-values, compute \(f(x) = ax^2 + bx + c\) for each, and plot the points to form a parabola. The vertex sits at \(x = -\frac{b}{2a}\), and the curve opens upward when \(a > 0\) and downward when \(a < 0\).
Which method should I use to solve a quadratic equation?
+Try factoring first, since it's the fastest when the numbers cooperate. If factoring doesn't work after a try or two, switch to the quadratic formula, which solves every quadratic without exception. Completing the square is worth using specifically when a question asks for vertex form.
Why can't 'a' be zero in a quadratic equation?
+If \(a\) were zero in the equation \(ax^2 + bx + c = 0\), the \(x^2\) term would become \(0 \cdot x^2 = 0\) and disappear. The equation would then be \(bx + c = 0\), which is a linear equation, not a quadratic one. The squared term is what gives a quadratic equation its unique properties and its parabolic graph.
Are roots and x-intercepts the same thing?
+For a quadratic written as \(y = ax^2 + bx + c\), real roots and \(x\)-intercepts describe the same \(x\)-values. A root solves \(ax^2 + bx + c = 0\), and an \(x\)-intercept is where the graph has \(y = 0\). Complex roots do not appear as intercepts on an ordinary coordinate graph.
What's the difference between a quadratic equation and a quadratic function?
+A quadratic function is the expression \(f(x) = ax^2 + bx + c\), describing a relationship between x and y. A quadratic equation sets that expression equal to zero and asks you to solve for x. The function describes the whole parabola; the equation focuses on where it crosses the x-axis.
Can a quadratic equation solver show complex answers?
+Yes, when the discriminant is negative. A quadratic calculator returns answers using \(i\).