Quadratic Equation Solver Guide: The Quadratic Formula Explained

A quadratic equation has the form a x squared plus b x plus c equals zero, with a not equal to zero. Its graph is a parabola, and solving the equation means finding where that parabola crosses the horizontal axis. Projectile paths, profit-maximising prices, and area problems with one unknown side all reduce to this shape.

The Formula and What Each Part Does

The solutions are x equals negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a. Read it in two halves. Negative b over 2a is the axis of symmetry, the x value at the vertex. The square root term is the distance from that axis out to each root. If that distance is zero, both roots sit on the axis and coincide.

The Discriminant Tells You the Answer Type

The discriminant is b squared minus 4ac.

• Positive: two distinct real roots, so the parabola cuts the axis twice.
• Zero: one repeated real root, the parabola touches the axis at its vertex.
• Negative: no real roots, two complex conjugate roots, and the parabola never reaches the axis.

Calculating the discriminant first is worth the extra line, because it tells you whether to expect a tidy answer, a repeated root, or complex values before you commit to the arithmetic.

A Fully Worked Example

Solve 2x squared minus 7x plus 3 equals 0. Here a = 2, b = -7, c = 3. The discriminant is 49 minus 24, which is 25, a perfect square, so the roots will be rational. The square root of 25 is 5, so x equals (7 plus or minus 5) divided by 4, giving x = 3 and x = 0.5. Check by substituting: 2 times 9 minus 21 plus 3 equals 0, correct.

Now try x squared plus 4x plus 7 equals 0. The discriminant is 16 minus 28 = -12, so there are no real roots. The complex solutions are -2 plus or minus the square root of 3 times i.

Factoring When It Is Faster

If the discriminant is a perfect square, factoring usually beats the formula. For x squared minus 5x plus 6, look for two numbers that multiply to 6 and add to -5: -2 and -3. So the equation factors to (x - 2)(x - 3) = 0 and the roots are 2 and 3. With a leading coefficient above 1, the factor pairs get fiddly quickly, and that is the point at which the formula is simply the faster tool.

Completing the Square and the Vertex

Rewriting a quadratic in vertex form gives you the turning point directly. For y equals x squared minus 6x plus 5, take half of -6 to get -3, square it to get 9, and write y equals (x - 3) squared minus 4. The vertex is at (3, -4), so the minimum value of the function is -4. This form is what you want when the question asks for a maximum profit, a minimum cost, or the peak height of a throw rather than the roots.

Reading the Roots From a Real Problem

A ball thrown upward at 20 m/s from a height of 1.5 m has height h = -4.9t squared + 20t + 1.5. Setting h to zero and solving gives t = -0.074 and t = 4.16. The negative root is mathematically valid but physically meaningless, so the ball lands after about 4.16 seconds. Discarding an impossible root is a normal part of applied work; always check whether a negative length, time, or price makes sense in the context.

The vertex gives the peak: t = -b / 2a = 20 / 9.8 = 2.04 s, and substituting back gives a maximum height of about 21.9 m.

Useful Relationships Between the Roots

For any quadratic, the sum of the roots is negative b over a and the product is c over a. In 2x squared minus 7x plus 3, the roots 3 and 0.5 sum to 3.5, which is 7/2, and multiply to 1.5, which is 3/2. This is a fast way to check an answer without resubstituting, and it lets you reconstruct an equation when you know only its roots.

Errors That Cost Marks

Sign errors on b are the most frequent, particularly forgetting that negative b of -7 is +7. Second is squaring b incorrectly when b is negative, since -7 squared is 49, not -49. Third is dividing only the square root term by 2a instead of the whole numerator. Writing the formula out with brackets before substituting prevents all three.

Solve Quadratics Instantly

Our free equation solver on SimpleCounter takes a, b, and c, reports the discriminant, gives real or complex roots, and shows the vertex, so you can check homework or a physics problem in seconds. It runs in the browser with no signup.