How do you solve an equation for x?
- 1Choose the kind of equation — or just type any equation in x.
- 2Type it with ^ for powers (the keys help on a phone), or fill in the coefficient boxes.
- 3Read the exact answer, its decimal value and the graph.
- 4Follow the step-by-step working to see how it's solved — and check your own homework.
The quadratic formula, discriminant and Cramer's rule
- 01Linear: ax + b = c → x = (c − b) / a
- 02Quadratic: ax² + bx + c = 0 → x = (−b ± √(b² − 4ac)) / 2a
- 03Discriminant D = b² − 4ac: D > 0 two real roots, D = 0 one, D < 0 two complex roots
- 04Vertex of y = ax² + bx + c: x = −b / 2a
- 052 × 2 system (Cramer): x = Dₓ / D, y = Dᵧ / D with D = a₁b₂ − a₂b₁
A linear equation is solved by doing the same thing to both sides: expand the brackets, move the x-terms to one side and the numbers to the other, then divide by the number in front of x. The answer is put back into the original equation as a check.
A quadratic is first put in standard form ax² + bx + c = 0 with whole-number coefficients. The discriminant b² − 4ac decides how many real roots there are; the quadratic formula gives them exactly, with square roots simplified (√48 = 4√3) and complex roots written with i. When the roots are fractions the equation is also factorised. For cubics and higher, the rational root test finds fraction roots, synthetic division removes them, and what's left is solved as a quadratic — or numerically if nothing factors.
Systems of two equations are solved by elimination — multiply the equations so one unknown cancels, solve, then substitute back — and checked with Cramer's rule. Three equations use Gauss–Jordan row operations on the augmented matrix. If the equations contradict each other there is no solution; if they say the same thing there are infinitely many, written with a free letter t. All arithmetic is exact with fractions of any size.
Good to know
- Write powers with ^ — x^2 for x² — or use the x² key; 2x means 2 × x.
- Brackets matter: 3(x − 2) multiplies both terms inside by 3.
- If the discriminant is a perfect square (like 49), the quadratic factorises neatly.
- For word problems, write one equation per fact, then use the 2- or 3-equation mode.
Solving equations — questions students ask
How do you solve a quadratic equation?
Put it in the form ax² + bx + c = 0, work out the discriminant D = b² − 4ac, then use x = (−b ± √D) / 2a. For 2x² − 3x − 5 = 0: D = 9 + 40 = 49, so x = (3 ± 7) / 4, giving x = 5/2 or x = −1.
What does the discriminant tell you?
If b² − 4ac is positive there are two different real roots, if it is zero there is one repeated root, and if it is negative there are no real roots — the roots are complex numbers and the parabola doesn't cross the x-axis.
How do you solve two equations with two unknowns?
Make the coefficients of one unknown equal by multiplying the equations, add or subtract them so it cancels, solve for the other unknown, then substitute back. For 2x + 3y = 7 and x − y = 1: double the second, subtract, 5y = 5, so y = 1 and x = 2.
When does a system of equations have no solution?
When the equations contradict each other — for two unknowns, when the lines are parallel, like x + y = 2 and 2x + 2y = 5. If one equation is just a multiple of the other, they are the same line and there are infinitely many solutions.
Can it give exact answers instead of decimals?
Yes. Answers are fractions and simplified square roots, such as (3 + √41)/4 or 2 ± √3, with the decimal value next to them.
How do you solve a cubic equation?
Look for a simple root with the rational root test (try factors of the constant over factors of the leading coefficient), divide it out with synthetic division, and solve the quadratic that is left. x³ − 6x² + 11x − 6 = 0 has roots 1, 2 and 3.
Sources
- Quadratic formula and discriminant
- Rational root theorem
- Cramer's rule
- Khan Academy – Solving equations and systems of equations
Formulas last reviewed: September 30, 2026 ·How we check our formulas