How to Solve Linear and Quadratic Equations

The quadratic formula is the piece of school mathematics most adults half-remember: something with a minus b, something with a square root, something divided by 2a. The half that gets forgotten is the part under the root, which decides whether there is an answer at all.

What the Equation Solver does

Solves ax² + bx + c = 0 for x, giving both roots along with the discriminant that determines what kind of roots they are.

The discriminant, b² − 4ac, is the number worth reading first. Positive means two real roots, zero means one repeated root, negative means the parabola never touches the axis and the roots are complex.

Open the Equation Solver and follow the settings and checks below.

What the numbers mean

Setting	What it means
Formula	x = (−b ± √(b² − 4ac)) ÷ 2a
Discriminant > 0	Two distinct real roots — the curve crosses the x-axis twice
Discriminant = 0	One repeated root — the curve touches the axis and turns
Discriminant < 0	No real roots — the curve misses the axis entirely
a = 0	Not a quadratic; it collapses to the straight line bx + c = 0
Sum and product	Roots always sum to −b/a and multiply to c/a — a quick sanity check

How to use it

1. Rearrange your equation into the form ax² + bx + c = 0 first. Everything must be on one side.
2. Enter a, b and c with their signs — a missing minus is the usual cause of a wrong answer.
3. Read the discriminant before the roots.
4. Check your answer: the two roots should add to −b/a.

Practical advice

Always rearrange before entering anything. x² + 2x = 8 is not a = 1, b = 2, c = 8 — the eight has to move across, giving c = −8, and the sign flip changes the roots completely. This single step accounts for most wrong answers people get from a quadratic solver, because the equation as written looks close enough to the standard form to type straight in.

Common questions

What does a negative discriminant mean?

That the parabola never crosses the x-axis, so there is no real value of x that satisfies the equation. The roots still exist as complex numbers involving the square root of a negative, which matters in electrical engineering and signal processing but usually means "no solution" in a practical context.

What if a is zero?

Then it is not a quadratic at all — the x² term vanishes and you are left with the straight line bx + c = 0, which has the single solution x = −c/b. Dividing by 2a in the formula would also mean dividing by zero, which is why the equation type has to change rather than the arithmetic being pushed through.

How do I get my equation into the right form?

Move everything to one side so the other side is zero, then collect like terms. x² + 2x = 8 becomes x² + 2x − 8 = 0, giving a = 1, b = 2, c = −8. Skipping this and typing the numbers as they appear is the most common way to get a confidently wrong pair of roots.

Is there a quick way to check the answer?

Yes, and it is worth doing. The two roots always sum to −b/a and multiply to c/a. For x² + 2x − 8 = 0 the roots are 2 and −4: they sum to −2, which is −b/a, and multiply to −8, which is c/a. Both checks passing means the arithmetic is almost certainly right.

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