Quadratic Formula Calculator

What the quadratic formula solves

A quadratic equation is any equation that can be rearranged into the form ax^2 + bx + c = 0, with a not equal to zero. The x^2 term is what separates it from a straight line. The letters a, b and c are coefficients: fixed numbers that belong to one particular equation, not variables you are solving for.

The formula gives the roots, meaning the values of x that make the left side equal zero: x = (-b +/- sqrt(b^2 - 4ac)) / 2a. The plus or minus sign is the whole point. It produces two answers from one expression, because a parabola can cross the x axis in two places.

The piece under the square root, b^2 - 4ac, is called the discriminant. Work it out first and you know what kind of answer to expect before doing any more arithmetic.

Enter the three numbers above and the calculator returns the discriminant, both roots, the vertex, the sum and product of the roots, and the factored form. The check line puts a root back into the original expression, which catches sign errors faster than rereading your work.

Worked example: 2x^2 - 4x - 6 = 0

Here a = 2, b = -4 and c = -6. The discriminant comes first: b^2 - 4ac = (-4)^2 - 4 x 2 x (-6) = 16 + 48 = 64. The negative c matters twice, because subtracting the product of two negatives adds 48 instead of taking it away.

Two cheap checks confirm the pair. The sum of the roots should equal -b / a = 4 / 2 = 2, and 3 + (-1) = 2. The product should equal c / a = -6 / 2 = -3, and 3 x (-1) = -3. If either one is off, a sign went missing along the way.

The vertex sits at x = -b / 2a = 1, halfway between the roots. Its height is 2 x 1^2 - 4 x 1 - 6 = -8, which is the lowest point on the curve. Expand 2(x - 1)^2 - 8 and you get 2x^2 - 4x + 2 - 8, the original expression, so the factored form 2(x - 3)(x + 1) is the same equation written a third way.

A projectile: when does the ball land

In feet and seconds, the height of a thrown object is h = -16t^2 + v0 t + h0. The 16 is half of the 32 ft/s^2 pull of gravity, and that factor of one half is exactly why the formula has a square in it.

Take a ball released 3 ft above the ground at 40 ft/s. Its height is h = -16t^2 + 40t + 3, and landing means h = 0. Multiply by -1 to make the leading term positive: 16t^2 - 40t - 3 = 0.

The discriminant is (-40)^2 - 4 x 16 x (-3) = 1600 + 192 = 1792, and sqrt(1792) is about 42.332. The roots are t = (40 + 42.332) / 32 = 2.573 seconds and t = (40 - 42.332) / 32 = -0.073 seconds.

The negative root is not a mistake. It is the moment on the same arc when the ball would have been at ground level before the throw started. Throwing it out is a decision about the physical situation, not about the algebra, and it is worth saying so out loud when you show your work.

The top of the flight comes at t = 40 / 32 = 1.25 seconds, where the height is -16 x 1.25^2 + 40 x 1.25 + 3 = -25 + 50 + 3 = 28 ft. In metric units the same model reads -4.9t^2 + v0 t + h0, and feeding 16 into a metric problem is one of the most common homework errors there is.

The discriminant tells you what to expect

One line of arithmetic sorts every quadratic into three groups. That is worth doing before you reach for a calculator, because it tells you whether a decimal answer even exists.

In the negative case the square root turns imaginary: sqrt(-16) = 4i, so x = (-2 +/- 4i) / 2 = -1 +/- 2i. The graph of that equation never reaches the x axis. Its lowest point is (-1, 4), four units above zero, which is the visual version of the same fact.

When the discriminant is a perfect square, the roots come out as whole numbers or simple fractions, which is why textbook problems tend to have those values. A number like 1792 has no clean square root, so the answers are decimals that never terminate. The calculator shows six decimal places and the check line confirms the substitution still lands on zero.

Vertex, axis of symmetry and the factored form

The same parabola can be written three ways, and each one shows a different feature at a glance.

The vertex coordinates come straight from the coefficients: h = -b / 2a and k = c - b^2 / (4a). For this equation h = 4 / 4 = 1 and k = -6 - 16 / 8 = -8, which matches the graph.

The axis of symmetry is the vertical line x = h. Every point on the parabola has a mirror twin on the other side of that line, so the two roots always sit the same distance from it. Here the roots 3 and -1 are two units either side of x = 1.

Multiplying the factored form back out is the check that the three forms agree: 2(x - 3)(x + 1) = 2(x^2 - 2x - 3) = 2x^2 - 4x - 6. If the middle term does not come back, one of the factors has the wrong sign.

One caveat about the vertex form: it is exact only when the coefficients are exact. With roots like 2.572876 the factored form carries a rounding error, which is why the calculator labels it as an approximation rather than pretending the decimals are clean.

Vieta's rules: a thirty second check

For any quadratic with roots r1 and r2, the sum of the roots is -b / a and the product is c / a. Both follow from expanding a(x - r1)(x - r2) and comparing coefficients, and both are faster than redoing the formula.

Example: x^2 + 5x - 84 = 0 has roots 7 and -12. The sum is -5, and -b / a = -5. The product is -84, and c / a = -84. Two lines, both consistent, and no square root needed.

The product rule also finishes a problem when you already know one root. If 7 solves x^2 + 5x - 84 = 0, the other root must satisfy 7 x r2 = -84, so r2 = -12. That trick is useful on tests with multiple choice answers and in code that checks a solver against known inputs.

Vieta's rules do not replace the formula, because they do not tell you the roots in the first place. They are a verification step, and a good one: a sign error in b or c almost always breaks one of the two sums.

Word problems that turn into quadratics

Most real uses of the formula arrive as a sentence, not as an equation. Two patterns cover a large share of them.

A rectangle with a fixed area. A garden is 5 ft longer than it is wide and covers 84 sq ft. Write the width as x, the length as x + 5, and the area as x(x + 5) = 84. Expanding gives x^2 + 5x - 84 = 0. The discriminant is 25 + 336 = 361, sqrt(361) = 19, and x = (-5 + 19) / 2 = 7. The garden is 7 ft by 12 ft, which multiplies to 84. The other root, -12, is a length that cannot exist, so it is dropped.

Profit that peaks and falls. A small seller models monthly profit as P = -0.5x^2 + 40x - 600, where x is the number of units sold. Break-even means P = 0, so 0.5x^2 - 40x + 600 = 0, and the discriminant is 1600 - 1200 = 400. The roots are (40 + 20) / 1 = 60 and (40 - 20) / 1 = 20, so the seller loses money below 20 units and above 60. The best month is at x = 40, where profit is -0.5 x 1600 + 1600 - 600 = 200 dollars.

The pattern is always the same: build the equation, put everything on one side, then read the discriminant. What changes is which root makes sense in the world, and that judgment never comes from the formula.

Mistakes that change the answer

Almost every wrong answer from this formula traces back to one of six slips.

A seventh issue is not arithmetic at all: keeping a root that the situation forbids. A negative width, a negative time before the throw, a negative number of units. The algebra hands you two answers and the context decides which one is real, so state the reason when you discard one.

For the underlying rules, the OpenStax chapter on quadratic equations walks through the same steps with exercises, and MathWorld's entry on the quadratic equation covers the derivation and the complex case. Both use the same convention this page does: the standard form ax^2 + bx + c = 0 with a not equal to zero.