Free quadratic equation calculator. Enter coefficients a, b and c to find the two real roots of a quadratic equation step by step.
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Write your equation in standard form, ax² + bx + c = 0, and enter the three coefficients a, b and c. This quadratic equation solver first works out the discriminant, b² - 4ac, then applies the quadratic formula to find the roots. The plus sign gives the larger root and the minus sign gives the smaller root. This calculator covers real roots: when the discriminant is positive you get two distinct real roots, when it is zero you get one repeated real root, and when it is negative there are no real roots to report. Use the step by step examples below to follow every stage of the working.
A quadratic equation is an equation that can be written in standard form as ax² + bx + c = 0, where a, b and c are numbers and a is not zero. The letter a is the coefficient of x², b is the coefficient of x, and c is the constant term. If a were zero, the x² term would disappear and the equation would be linear, not quadratic, which is why this quadratic equation calculator needs a nonzero value for a. The solutions of the equation, called its roots, are the values of × that make the equation true. You can also write the same equation in the compact form ax2 + bx + c = 0, which is how many textbooks and the ax2 bx c calculator tools on the web show it.
The quadratic formula solves every quadratic equation, no matter how awkward the coefficients are:
roots = (-b +/- sqrt(b² - 4ac)) / (2a)
The symbol +/- means you do the calculation twice: once with a plus sign to get one root, and once with a minus sign to get the other root. That is why a quadratic equation can have two solutions. The part under the square root, b² - 4ac, is called the discriminant, and it decides the type of roots before you finish the calculation. The formula works for whole numbers, decimals and negative coefficients alike, and it never fails for a genuine quadratic equation, which is why it is the most reliable way to solve quadratic equation problems by hand or with a quadratic formula solver.
The discriminant is the expression b² - 4ac from inside the quadratic formula. Work it out first, because it tells you how many real roots to expect:
| Discriminant | Roots | Graph | |---|---|---| | Positive | Two distinct real roots | Parabola crosses the x-axis twice | | Zero | One repeated real root | Parabola touches the x-axis at the vertex | | Negative | No real roots, two complex roots | Parabola never meets the x-axis |
This page is a quadratic equation calculator discriminant tool as well as a root finder: the sign of the discriminant is the first thing to check in every worked example below. A discriminant that is a perfect square, such as 1, 4, 9 or 25, usually means the roots are simple whole numbers or fractions, which is a useful hint when you want to factor the equation instead.
Follow these steps to solve quadratic equation problems with this calculator:
1. Write your equation in standard form, ax² + bx + c = 0. Move every term to one side if needed. 2. Read off the coefficients: a is the number in front of x², b is the number in front of x, and c is the constant term. Watch the signs, a missing number means a coefficient of 1, and a missing term means a coefficient of 0. 3. Enter a, b and c in the three boxes above. Decimals and negative numbers are accepted. 4. Read the two results: the larger root and the smaller root, each rounded to two decimal places. 5. Check your answers by putting each root back into the original equation. Each one should give zero, or very close to zero after rounding.
This quadratic equation calculator with steps approach mirrors the working shown in the examples below, so you can use it for homework, revision or a quick check of hand working.
Example 1, two real roots: solve x² - 5x + 6 = 0. Here a = 1, b = -5 and c = 6. The discriminant is (-5)² - 4 × 1 × 6 = 25 - 24 = 1. The roots are (5 +/- sqrt(1)) / 2, which gives (5 + 1) / 2 = 3 and (5 - 1) / 2 = 2. Check: 3 + 2 = 5 and 3 × 2 = 6, matching the original coefficients.
Example 2, a repeated root: solve x² - 3x + 2 = 0. Here a = 1, b = -3 and c = 2. The discriminant is (-3)² - 4 × 1 × 2 = 9 - 8 = 1, so the roots are (3 +/- 1) / 2, giving 2 and 1.
Example 3, irrational roots: solve x² - 8x + 5 = 0. The discriminant is 64 - 20 = 44, so the roots are 4 +/- sqrt(11), about 7.32 and 0.68. These are real roots even though they are not whole numbers.
Example 4, no real roots: solve x² + 2x + 5 = 0. The discriminant is 4 - 20 = -16, which is negative, so there are no real roots. A negative discriminant means the parabola never crosses the x-axis, and the two solutions are complex numbers, which this real root calculator does not report.
There are three main ways to solve a quadratic equation, and each suits a different situation:
| Method | Best when | Drawback | |---|---|---| | Factoring | The roots are whole numbers or simple fractions, and the discriminant is a perfect square | Fails or gets messy for irrational roots | | Completing the square | You need the vertex form, or you want to see where the formula comes from | Slow and error prone with fractions | | Quadratic formula | Always, especially with decimals, large coefficients or irrational roots | Gives decimal answers rather than exact surds unless simplified |
When to use quadratic formula vs factoring: try factoring first if the equation looks friendly, for example x² - 5x + 6 = 0 factors into (x - 3)(x - 2) = 0. Reach for the quadratic formula whenever factoring is not obvious, because it always works. Completing the square vs quadratic formula is not really a contest: completing the square is the derivation of the formula itself, so learning it once explains why the formula works, and then the formula does the heavy lifting for everyday problems.
Every quadratic equation is linked to a quadratic equation graph parabola, the curve y = ax² + bx + c. The roots are exactly the points where that parabola crosses the x-axis, which is why a positive discriminant gives two crossings, a zero discriminant gives one touching point, and a negative discriminant gives no crossing at all. The vertex of the parabola, its highest or lowest point, sits at × = -b / (2a), exactly halfway between the two roots when they are real. The vertical line through the vertex, × = -b / (2a), is the axis of symmetry. If a is positive the parabola opens upwards and the vertex is a minimum; if a is negative it opens downwards and the vertex is a maximum. This picture gives a fast visual check: if your calculated roots do not sit symmetrically around -b / (2a), something went wrong.
Quadratic equations show up far beyond the classroom, which is why a quadratic equation solver is useful for students, engineers and analysts alike:
- Projectile motion: the height of a thrown ball or a rocket follows a quadratic curve, so finding when it hits the ground means solving a quadratic equation. - Engineering and design: optimisation problems, such as finding the dimensions that give the largest area for a fixed perimeter, lead to quadratic equations. - Business and profit models: revenue and profit often follow a parabolic shape, and the break even points are the roots of a quadratic equation. - Area and construction problems: finding the side of a square or rectangular plot from its area and perimeter difference is a classic quadratic equation real life uses example. - Science and trajectory modelling: satellite paths, lens design and signal processing all use quadratic relationships.
In each case the same steps apply: write the equation in standard form, identify a, b and c, work out the discriminant, and apply the quadratic formula.
The quadratic formula is roots = (-b +/- sqrt(b² - 4ac)) / (2a). It solves any quadratic equation written in standard form, ax² + bx + c = 0, where a is not zero. The plus sign gives one root and the minus sign gives the other root.
The discriminant is b² - 4ac, the part under the square root in the quadratic formula. A positive discriminant means two distinct real roots, a zero discriminant means one repeated real root, and a negative discriminant means no real roots, only two complex roots.
A negative discriminant means the equation has no real roots, because a negative number has no real square root. On a graph, the parabola never crosses the x-axis. This calculator covers real roots, so for a negative discriminant it reports no real root result, and the two solutions are complex numbers of the form p plus/minus qi.
First write the equation in standard form, ax² + bx + c = 0, and read off a, b and c. Then work out the discriminant, b² - 4ac. Finally apply the quadratic formula, roots = (-b +/- sqrt(b² - 4ac)) / (2a), doing the calculation once with a plus sign and once with a minus sign. You can also find roots by factoring when the equation factors neatly, or by completing the square.
Step 1: write the equation in standard form, ax² + bx + c = 0. Step 2: identify the coefficients a, b and c, keeping their signs. Step 3: calculate the discriminant, b² - 4ac. Step 4: put the numbers into the quadratic formula. Step 5: simplify the square root, then take the plus and minus branches separately to get the two roots. Step 6: check each root by substituting it back into the original equation, which should give zero.
Use factoring when the roots are obvious whole numbers or simple fractions, for example x² - 5x + 6 = 0, which factors into (x - 3)(x - 2) = 0. Use the quadratic formula whenever factoring is not obvious, when coefficients are decimals, or when the roots are irrational. The formula always works, so it is the safe default for homework and exams.
To factor a quadratic equation, look for two numbers that add up to b and multiply to c, for an equation of the form x² + bx + c = 0. For x² - 5x + 6 = 0, the numbers are -3 and -2, because -3 + -2 = -5 and -3 × -2 = 6, so the equation factors into (x - 3)(x - 2) = 0, giving roots 3 and 2. If no such pair exists, use the quadratic formula instead.
Standard form of a quadratic equation is ax² + bx + c = 0, where a, b and c are numbers and a is not zero. Every quadratic equation can be rearranged into this form by moving all terms to one side, which is the form the quadratic formula expects.
If a = 0, the x² term disappears and the equation becomes linear, bx + c = 0, with the single solution × = -c / b. That is not a quadratic equation, and the quadratic formula cannot be used because it divides by 2a. This calculator requires a nonzero value for a.
The vertex of the parabola y = ax² + bx + c sits at × = -b / (2a), which is exactly halfway between the two roots when they are real. To find the full vertex point, calculate × = -b / (2a), then substitute that × value back into ax² + bx + c to get the y coordinate. The line × = -b / (2a) is the axis of symmetry.
Yes. Enter any real numbers for a, b and c, including decimals and negative values, as long as a is not zero and each value stays within the input range. The roots are rounded to two decimal places.
Substitute each root back into the original equation: it should give zero, or very close to zero after rounding. You can also check with the sum and product rules: the two roots add up to -b / a and multiply to c / a. For x² - 5x + 6 = 0, the roots 3 and 2 add to 5 and multiply to 6, which matches.
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