What is the discriminant and why is it important?

The discriminant is b²-4ac in the quadratic formula. It determines the nature of solutions: Positive discriminant means two real different solutions. Zero discriminant means one repeated real solution. Negative discriminant means two complex conjugate solutions. It helps predict solution types before calculating.

Is this solver suitable for GCSE and A-Level maths?

Yes! This equation solver covers all linear and quadratic equations taught in GCSE and A-Level maths. It shows step-by-step working, explains discriminant, and handles real and complex solutions - perfect for homework checking and exam preparation.

Equation Solver UK - Linear & Quadratic Equations

Last verified: 2026-01-12 • Updated for 2026/26 tax year

Solve linear and quadratic equations online. Perfect for GCSE and A-Level maths students. Get instant solutions with step-by-step explanations. Updated 2025.

Solve Your Equation

Equation Type

a (coefficient of x)

b (constant)

📚 Worked Examples

Example 1: Solving a Linear Equation

Problem: Solve 3x - 9 = 0

Step 1: Identify coefficients
a = 3, b = -9

Step 2: Apply formula x = -b/a
x = -(-9)/3 = 9/3 = 3

Step 3: Check solution
3(3) - 9 = 9 - 9 = 0 ✓

Answer: x = 3

Example 2: Quadratic with Two Real Solutions

Problem: Solve x² - 5x + 6 = 0

Step 1: Identify coefficients
a = 1, b = -5, c = 6

Step 2: Calculate discriminant
Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1

Step 3: Since Δ > 0, two real solutions exist
x = (-b ± √Δ) / 2a = (5 ± 1) / 2

Step 4: Calculate both solutions
x₁ = (5 + 1) / 2 = 3
x₂ = (5 - 1) / 2 = 2

Answer: x = 3 or x = 2

Example 3: Quadratic with One Solution (Perfect Square)

Problem: Solve x² - 6x + 9 = 0

Step 1: Identify coefficients
a = 1, b = -6, c = 9

Step 2: Calculate discriminant
Δ = (-6)² - 4(1)(9) = 36 - 36 = 0

Step 3: Since Δ = 0, one repeated solution
x = -b / 2a = 6 / 2 = 3

Step 4: Recognize as perfect square
(x - 3)² = 0, so x = 3

Answer: x = 3 (repeated root)

Example 4: Quadratic with Complex Solutions

Problem: Solve x² + 2x + 5 = 0

Step 1: Identify coefficients
a = 1, b = 2, c = 5

Step 2: Calculate discriminant
Δ = (2)² - 4(1)(5) = 4 - 20 = -16

Step 3: Since Δ < 0, complex solutions
√(-16) = 4i (where i = √-1)

Step 4: Apply quadratic formula
x = (-2 ± 4i) / 2 = -1 ± 2i

Answer: x = -1 + 2i or x = -1 - 2i

Example 5: Rearranging Before Solving

Problem: Solve 2x² + 3x = 5

Step 1: Rearrange to standard form
2x² + 3x - 5 = 0

Step 2: Identify coefficients
a = 2, b = 3, c = -5

Step 3: Calculate discriminant
Δ = 9 - 4(2)(-5) = 9 + 40 = 49

Step 4: Solve using quadratic formula
x = (-3 ± 7) / 4
x₁ = 4/4 = 1
x₂ = -10/4 = -2.5

Answer: x = 1 or x = -2.5

💡 Tips for Solving Equations

Linear Equations

Isolate the variable: Get x by itself on one side
Inverse operations: Add to remove subtraction, divide to remove multiplication
Check your work: Substitute solution back into original equation
Special cases: If a=0, equation may have no solution or infinite solutions

Quadratic Equations

Standard form first: Always rearrange to ax² + bx + c = 0
Check discriminant: Tells you solution type before calculating
Factoring shortcut: If equation factors easily, use that instead
Completing the square: Alternative method, especially for perfect squares
Sum and product: x₁ + x₂ = -b/a, x₁ × x₂ = c/a (useful for checking)

Common Shortcuts

Recognize patterns: (x±a)² = x² ± 2ax + a²
Difference of squares: x² - a² = (x+a)(x-a)
Calculator check: Use calculator to verify, but show working
Graph visualization: Solutions are x-intercepts

⚠️ Common Mistakes to Avoid

❌ Mistake 1: Sign errors in quadratic formula

WRONG: Forgetting the negative in -b

CORRECT: x = (-b ± √Δ) / 2a, not (b ± √Δ) / 2a

❌ Mistake 2: Forgetting both solutions

WRONG: Only writing x = (-b + √Δ) / 2a

CORRECT: Write both x₁ and x₂ using ± symbol

❌ Mistake 3: Incorrect discriminant

WRONG: Δ = b² - 4c (forgetting a)

CORRECT: Δ = b² - 4ac (include the a!)

❌ Mistake 4: Not rearranging to standard form

WRONG: Trying to solve 2x² = 8 - 3x directly

CORRECT: Rearrange to 2x² + 3x - 8 = 0 first

❌ Mistake 5: Division errors

WRONG: (-b ± √Δ) / 2a = -b/2a ± √Δ/2a

CORRECT: Divide ENTIRE numerator by 2a

📖 Complete Guide to Equation Solving

Understanding Equations

An equation is a mathematical statement that two expressions are equal. Solving an equation means finding all values of the variable that make the equation true. Linear equations have degree 1 (highest power is x¹), while quadratic equations have degree 2 (highest power is x²).

The Quadratic Formula

For any quadratic equation ax² + bx + c = 0 (where a ≠ 0):
x = (-b ± √(b² - 4ac)) / (2a)

This formula works for ALL quadratic equations and is derived by completing the square. Memorize it - it's essential for GCSE and A-Level maths!

The Discriminant Explained

The discriminant Δ = b² - 4ac tells you about solutions BEFORE calculating:

Δ > 0: Two different real solutions (parabola crosses x-axis twice)
Δ = 0: One repeated real solution (parabola touches x-axis once)
Δ < 0: Two complex solutions (parabola doesn't cross x-axis)

Methods for Solving Quadratics

Factoring: Write as (x+p)(x+q)=0, solve x=-p or x=-q. Fast but only works for nice numbers.
Quadratic Formula: Always works! Use when factoring is difficult.
Completing the Square: Rewrite as (x+h)²=k. Good for deriving vertex form.
Graphing: Find x-intercepts. Useful for visualization but not always accurate.

Real-World Applications

Projectile motion: Height equations are quadratic (h = -5t² + vt + h₀)
Area problems: Finding dimensions when area is known
Economics: Profit maximization, break-even analysis
Physics: Acceleration, energy equations
Engineering: Structural design, optimization

❓ Frequently Asked Questions

How do you solve a quadratic equation?

Use the quadratic formula: x = (-b ± √(b²-4ac)) / (2a). First calculate the discriminant (b²-4ac). If positive, you get two real solutions. If zero, one solution. If negative, two complex solutions. Always show your working and check solutions by substituting back.

What is a linear equation?

A linear equation has the form ax + b = 0, where a and b are constants and a ≠ 0. Solve by rearranging: x = -b/a. Linear equations always have exactly one solution and graph as straight lines.

What are complex solutions?

Complex solutions occur when the discriminant is negative. They involve imaginary numbers (i, where i² = -1). Written as a + bi, where a is the real part and b is the imaginary part. Complex solutions always come in conjugate pairs for equations with real coefficients.

Can this solver help with GCSE maths?

Yes! This equation solver is perfect for GCSE and A-Level students. It solves quadratic equations taught in GCSE maths and shows the discriminant, which is important for understanding solution types. Use it to check homework and understand step-by-step solutions.

What if coefficient 'a' is zero?

If a = 0 in a quadratic equation, it becomes linear (bx + c = 0). If a = 0 in a linear equation, the equation has no solution (if b ≠ 0) or infinitely many solutions (if b = 0 too).

How do I check my answer is correct?

Substitute your solution(s) back into the original equation. If the left side equals the right side, your answer is correct. For quadratic equations, check BOTH solutions if there are two.

Is this calculator free?

Yes! Completely free with no signup required. Unlimited calculations for homework, revision, and exam preparation. Works on phones, tablets, and computers.

What's the difference between roots, solutions, and zeros?

These terms mean the same thing! "Solutions" and "roots" refer to the values of x that satisfy the equation. "Zeros" refers to where the graph crosses the x-axis (y=0). All three terms are used interchangeably in maths.