Line Slope Finder

Quadratic Equation Solver

Solve any quadratic equation and see every step. Type the equation the way your question writes it, such as x² − x − 6 = 0, (x + 2)(x − 3) = 0 or x² = x + 6, then choose how the working is shown:

  • Factorising: finds the brackets and sets each one equal to zero, and tells you when an equation doesn’t factorise.
  • Quadratic formula: substitutes into x = (−b ± √(b² − 4ac)) ÷ 2a. It always works and gives exact surd answers.
  • Intersecting lines: rearranges to x² = mx + k and finds where the line crosses the parabola y = x² on the graph.
Or set a, b and c for ax² + bx + c = 0 x² + x + = 0
y = x² parabolay = x + 6 line

Drag point P or Q to move the line and watch where it meets the parabola. The equation above changes to match.

Solutions

Working

How to use the Quadratic Solver

Entering an equation

  • Type the equation exactly as your question gives it. These all work: x^2 - x - 6 = 0, (x + 2)(x - 3) = 0, x² = x + 6, 2(x - 1)^2 = 8. Use ^2 or ² for squared.
  • Or write it as ax² + bx + c = 0 and type a, b and c into the boxes. For x² − x − 6 = 0, type 1, -1 and -6.
  • Fractions such as 1/2 and decimals such as 0.5 both work. a can’t be 0, because then the equation isn’t quadratic.
  • The example buttons show each case: two roots, a not equal to 1, surd roots, one repeated root, and no real roots.

Using the graph

  • The green curve is always y = x². The blue line comes from your equation.
  • Red rings mark where the line crosses the parabola. Their x values are the solutions.
  • Drag P or Q to move the line. The a, b and c boxes update, so you can see which equation each line solves.
  • The graph zooms itself so the crossing points are always in view.

Reading the answer

  • Two solutions: the line cuts through the parabola at two points.
  • One repeated solution: the line just touches the parabola. It is a tangent.
  • No real solutions: the line passes underneath the parabola without meeting it.
  • Answers are exact. Roots that aren’t whole numbers or fractions are shown as surds, such as 1 ± √2, with decimals beside them.
  • Choose how the working is shown. Intersecting lines rearranges to x² = mx + k and reads the graph. Factorising finds the brackets, such as (x + 2)(x − 3) = 0, and sets each one to zero; it tells you when an equation doesn’t factorise. Quadratic formula substitutes into x = (−b ± √(b² − 4ac)) / 2a and always works.

Learn the methods

Practice problems 19 to 26 on the practice page cover all three methods.