Equation solver

Solve 2x² − 3x − 5 = 0.

1. a = 2, b = −3 and c = −5.
2. Discriminant: (−3)² − 4 × 2 × (−5) = 9 + 40 = 49.
3. √49 = 7, so x = (3 ± 7) / 4.
4. x = 10/4 = 5/2, or x = −4/4 = −1.

Answer: x = 5/2 (2.5) or x = −1.

• Losing the sign of b: with b = −3, −b is +3 and b² is +9, never −9.
• Dividing by 2 instead of 2a at the end of the quadratic formula.
• Reading off a, b and c before the equation equals 0: x² = 5x − 6 must become x² − 5x + 6 = 0 first.
• Cancelling x from both sides of x² = 3x, which loses the root x = 0. Factorise instead: x(x − 3) = 0.

This equation solver handles the three kinds of equation you meet most in school maths: linear equations in x, quadratics, and pairs of simultaneous equations. It does more than give the answer. You see the working, from the discriminant and the quadratic formula to each step of elimination, and answers come as exact fractions where possible, with decimals alongside. It also solves linear and quadratic inequalities with a number line, and expands, factorises and simplifies expressions, showing each step.

1. Pick Solve, Inequality, Expand, Factorise or Simplify at the top. Solve has Linear, Quadratic and Simultaneous.
2. Pick a mode: Linear, Quadratic or Simultaneous.
3. Linear: type the whole equation in x, for example 3(x - 2) = 2x + 5. The solver expands the brackets, collects terms and gives x = 11.
4. Quadratic: enter a, b and c from ax² + bx + c = 0. For x² - 5x + 6 = 0, that is a = 1, b = -5 and c = 6.
5. Simultaneous: enter the numbers for two equations in the form ax + by = c. The solver eliminates one variable, substitutes back and checks the answer in both equations.

• The discriminant b² - 4ac tells you how many real roots to expect: positive means two, zero means one, negative means none (the solver then shows the complex roots).
• When the roots are rational, the solver also shows the factorised form, for example (x - 2)(x - 3) for x² - 5x + 6. Use it to check your own factorising.
• The vertex (turning point) is useful for sketching: it sits at x = -b / 2a, halfway between the two roots.
• If simultaneous equations turn out to be parallel lines, there is no solution. If they are the same line, every point on it works. Both are valid answers, not mistakes.

What does "no real solutions" mean?

No real number makes the equation true: the graph of the quadratic never crosses the x-axis. It happens when the discriminant b² − 4ac is negative, because the square root in the quadratic formula then has no real value.

How do you solve simultaneous equations?

Make one unknown disappear. With elimination, scale the equations so one variable has the same coefficient in both, then add or subtract them. With substitution, rearrange one equation for one variable and put it into the other. Find one unknown, then substitute back to find the other.

What if my equation has no solution?

The linear mode tells you when there is no solution (for example x + 1 = x + 2) or when every value of x works. It also tells you if the equation is not linear, such as one with x², so you can switch to the Quadratic mode.

What happens if a = 0 in a quadratic?

Then there is no x² term, so it is not a quadratic. The solver treats it as the linear equation bx + c = 0 and solves that instead.

Why are answers shown as fractions?

Exact answers are what most mark schemes want. A fraction such as 2/3 is exact, while 0.6667 is rounded. The solver shows both so you can use whichever the question asks for.