Short answer
What you need to know
Put the equation in the form ax^2 + bx + c = 0. If there is no x term, take square roots. If it factors with small whole numbers, factor. Otherwise use the quadratic formula, and read the discriminant b^2 - 4ac first to know whether to expect two real, one repeated, or two complex roots.
Get to standard form first
Move every term to one side so the equation reads ax^2 + bx + c = 0. Then read off a, b, and c with their signs. In 3x^2 = 5x + 2, subtracting gives 3x^2 - 5x - 2 = 0, so a = 3, b = -5, and c = -2.
If every coefficient shares a factor, divide it out. 2x^2 - 8x + 6 = 0 becomes x^2 - 4x + 3 = 0, which is easier to work with and has the same solutions.
Method 1: square roots
When there is no x term, isolate x^2 and take the square root of both sides, remembering both signs. 4x^2 - 36 = 0 gives x^2 = 9, so x = 3 or x = -3.
Method 2: factoring
Look for two numbers that multiply to a·c and add to b. For x^2 - x - 12 = 0 you need two numbers with product -12 and sum -1: they are -4 and 3. So (x - 4)(x + 3) = 0 and x = 4 or x = -3.
Factoring is the fastest method when it works, but most quadratics do not factor with whole numbers. If you cannot find the pair in a minute, switch methods.
Method 3: completing the square
Move the constant to the other side, then add (b/2)^2 to both sides so the left side becomes a perfect square. For x^2 + 6x - 7 = 0: x^2 + 6x = 7, add 9 to get x^2 + 6x + 9 = 16, which is (x + 3)^2 = 16. So x + 3 = ±4, giving x = 1 or x = -7.
This method matters beyond solving. It is how you rewrite a quadratic in vertex form, a(x - h)^2 + k, and it is how the quadratic formula is derived.
Method 4: the quadratic formula
x = (-b ± sqrt(b^2 - 4ac)) / (2a) solves every quadratic. For 2x^2 + 4x - 3 = 0: b^2 - 4ac = 16 + 24 = 40, so x = (-4 ± sqrt(40)) / 4.
Simplify the radical: sqrt(40) = 2·sqrt(10), so x = (-2 ± sqrt(10)) / 2. As decimals, x ≈ 0.581 or x ≈ -2.581.
What the discriminant tells you
The discriminant D = b^2 - 4ac is the part under the square root. Work it out first.
D greater than 0: two different real roots, and the parabola crosses the x-axis twice. If D is a perfect square, the roots are rational and the quadratic factors.
D equal to 0: one repeated real root, and the parabola touches the axis at its vertex.
D less than 0: no real roots, two complex roots, and the parabola never meets the axis. For x^2 + 2x + 5 = 0, D = 4 - 20 = -16, so x = (-2 ± 4i) / 2 = -1 ± 2i.
Choosing a method
No x term: square roots. Small coefficients and a perfect-square discriminant: factoring. Asked for vertex form or the maximum or minimum: completing the square. Anything else, or when you are unsure: the formula.
Whatever the method, check one root by substituting it back. It takes ten seconds and catches most sign errors.
From equations to inequalities
To solve ax^2 + bx + c > 0 or < 0, find the roots first, then decide which side of them you want. When a is positive the parabola opens upward, so it is below the axis between the roots and above it outside them. For x^2 - x - 12 < 0 the roots are -3 and 4, and the solution is -3 < x < 4.
Calculator programs for this topic
These free TI-84 Plus CE Python programs carry out the steps above. Each page explains its own math, shows a screen preview, and lets you read the source before you download it.
- DuckieDai Quadratic Solver: Applies the formula, shows the discriminant, and gives real or complex roots.
- DuckieDai Quadratic Factor Finder: Finds the whole-number factors, or tells you the quadratic does not factor.
- DuckieDai Quadratic Inequality Solver: Solves quadratic inequalities and gives the answer in interval notation.
- DuckieDai Radical Simplifier: Simplifies the square root in a formula answer, such as sqrt(40) = 2·sqrt(10).
Try these before revealing the answers
Solve x^2 - 49 = 0.
Show worked answer
x^2 = 49, so x = 7 or x = -7.
Solve x^2 + 5x + 6 = 0 by factoring.
Show worked answer
2 and 3 multiply to 6 and add to 5, so (x + 2)(x + 3) = 0 and x = -2 or x = -3.
How many real roots does 3x^2 - 2x + 1 = 0 have?
Show worked answer
D = 4 - 12 = -8, which is negative, so none: two complex roots.
Important limitation
Check your answers
These methods solve quadratics with real coefficients. Decimal answers from a calculator are approximations, so give exact radical answers when your course asks for them, and check each root in the original equation.
Further reading
Authoritative references
These primary sources support the concepts and reference information in this guide. Their inclusion does not imply endorsement of Hidden Word Scanner.