Equation Rearranger & Solver UK 2026
Make x the subject, rearrange formulas, expand brackets & solve linear or quadratic equations step by step. Free algebraic equation calculator for GCSE and A-Level.
Last updated: February 2026
Equation Solver UK - Linear & Quadratic Equations
Solve linear and quadratic equations online. Perfect for GCSE and A-Level maths students. Get instant solutions with step-by-step explanations. Updated 2026.
Solve Your Equation
Worked Examples
Example 1: Solving a Linear Equation
Problem: Solve 3x - 9 = 0
a = 3, b = -9
x = -(-9)/3 = 9/3 = 3
3(3) - 9 = 9 - 9 = 0
Answer: x = 3
Example 2: Quadratic with Two Real Solutions
Problem: Solve x² - 5x + 6 = 0
a = 1, b = -5, c = 6
Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
x = (-b ± √Δ) / 2a = (5 ± 1) / 2
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
a = 1, b = -6, c = 9
Δ = (-6)² - 4(1)(9) = 36 - 36 = 0
x = -b / 2a = 6 / 2 = 3
(x - 3)² = 0, so x = 3
Answer: x = 3 (repeated root)
Example 4: Quadratic with Complex Solutions
Problem: Solve x² + 2x + 5 = 0
a = 1, b = 2, c = 5
Δ = (2)² - 4(1)(5) = 4 - 20 = -16
√(-16) = 4i (where i = √-1)
x = (-2 ± 4i) / 2 = -1 ± 2i
Answer: x = -1 + 2i or x = -1 - 2i
Example 5: Rearranging Before Solving
Problem: Solve 2x² + 3x = 5
2x² + 3x - 5 = 0
a = 2, b = 3, c = -5
Δ = 9 - 4(2)(-5) = 9 + 40 = 49
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.
Equation Rearranger: Make X the Subject
Rearranging an equation means isolating a particular variable on one side. The golden rule is: whatever you do to one side, you must do to the other side. This tool and guide covers the most common equation rearrangement techniques used in GCSE and A-Level maths, physics, and chemistry.
How to Make X the Subject — Step-by-Step Method
- Identify what you want to isolate — decide which variable you want to make the subject.
- Move all terms containing x to one side — use addition or subtraction to move other terms to the opposite side.
- Simplify — collect like terms and simplify coefficients.
- Divide or multiply — divide both sides by the coefficient of x to isolate it.
- Apply inverse operations — undo squares with square roots, undo logarithms with exponentials, etc.
- Check your answer — substitute back into the original equation to verify.
Worked Examples: Make X the Subject
Example 1: Make x the subject of y = mx + c
Example 2: Make x the subject of v = u + at (rearrange for a)
Example 3: Make x the subject of A = πr² (rearrange for r)
Example 4: Make x the subject of ax² + bx + c = 0 (quadratic formula)
Equation Rearranger Reference Table
Common physics, maths, and science formulas rearranged. Use these as a quick reference for GCSE and A-Level. Click the formula to copy.
| Formula Type | Original Formula | Rearranged For | Result |
|---|---|---|---|
| Linear | y = mx + c | x | x = (y−c)/m |
| Speed (SUVAT) | v = u + at | a | a = (v−u)/t |
| Speed (SUVAT) | v = u + at | t | t = (v−u)/a |
| Circle area | A = πr² | r | r = √(A/π) |
| Volume box | V = lwh | l | l = V/(wh) |
| Energy (E=mc²) | E = mc² | m | m = E/c² |
| Ohm's Law | V = IR | R | R = V/I |
| Ohm's Law | V = IR | I | I = V/R |
| Force | F = ma | m | m = F/a |
| Density | D = m/V | V | V = m/D |
| Pressure | P = F/A | A | A = F/P |
| Kinetic energy | KE = ½mv² | v | v = √(2KE/m) |
| Profit margin | P% = (P/C)×100 | C | C = P×100/P% |
Expand and Factorise Brackets
Expanding brackets is a fundamental algebraic skill. The key rule is to multiply each term inside the bracket by the term outside (or by each term in the other bracket). This section covers the most common types encountered in GCSE maths, including examples like expand 3(x+4) and (x+3)(x+4).
Single Bracket Expansion
Multiply every term inside the bracket by the factor outside:
Expand 3(x + 4)
Expand −2(x − 5)
Double Bracket Expansion (FOIL Method)
The FOIL method (First, Outer, Inner, Last) is the standard technique for multiplying two brackets:
- First — multiply the first terms of each bracket
- Outer — multiply the outer terms
- Inner — multiply the inner terms
- Last — multiply the last terms of each bracket
Expand (x + 3)(x + 4)
Expand (x + 3)(x − 5)
Common Expansion Patterns
| Expression | Expanded Form | Pattern Name |
|---|---|---|
| (x + a)² | x² + 2ax + a² | Perfect square (positive) |
| (x − a)² | x² − 2ax + a² | Perfect square (negative) |
| (x + a)(x − a) | x² − a² | Difference of two squares |
| (x + a)(x + b) | x² + (a+b)x + ab | General FOIL |
| a(b + c + d) | ab + ac + ad | Distributive law |
| (2x + 3)² | 4x² + 12x + 9 | Perfect square (numeric) |
| (x+3)(x+4)(x+5) | Expand two brackets first, then third | Triple bracket |
Equation Rearranger: Frequently Asked Questions
How do I rearrange a formula?
To rearrange a formula, perform the same inverse operation on both sides. To isolate x: move all terms not containing x to the other side using addition or subtraction, then divide (or multiply) both sides by x's coefficient. For squared terms, take the square root of both sides at the end. Always check your rearranged formula by substituting numbers back in.
What is the equation rearranger formula?
There is no single “rearranger formula” — different equation types require different techniques. For linear equations (y = mx + c), rearranging for x gives x = (y−c)/m. For quadratic equations, the quadratic formula x = (−b ± √(b²−4ac)) / (2a) is the universal rearrangement. The reference table above covers the most common formulas from physics, chemistry, and maths.
How do you make x the subject of a formula?
Making x the subject means isolating x on the left-hand side of the equation. The steps are: (1) add or subtract terms to collect all x terms on one side; (2) expand any brackets containing x; (3) factorise if x appears more than once; (4) divide both sides by x's coefficient. For example, to make x the subject of 3x + 7 = y: subtract 7 to get 3x = y−7, then divide by 3 to get x = (y−7)/3.
How do you expand brackets?
To expand brackets, multiply each term inside the bracket by the factor outside. For a single bracket: a(b + c) = ab + ac. For double brackets (FOIL method): (x+3)(x+4) = x² + 4x + 3x + 12 = x² + 7x + 12. Remember to pay attention to signs — a negative multiplied by a negative gives a positive.
What does x² + 5x + 6 factorise to?
x² + 5x + 6 factorises to (x + 2)(x + 3). To check: expand using FOIL: x² + 3x + 2x + 6 = x² + 5x + 6. When factorising a quadratic x² + bx + c, find two numbers that multiply to c and add to b. Here, 2 × 3 = 6 and 2 + 3 = 5.
How do you solve x² + 3x − 5 = 0 (written as “x 3 x 5 2”)?
For x² + 3x − 5 = 0, use the quadratic formula with a=1, b=3, c=−5. Discriminant = 3² − 4(1)(−5) = 9 + 20 = 29. Solutions: x = (−3 ± √29) / 2. This gives x ≈ 1.193 or x ≈ −4.193. Use our equation solver above to calculate this instantly with full step-by-step working.
What is an algebraic equation calculator used for?
An algebraic equation calculator solves equations for an unknown variable. Common uses include: solving linear equations (ax + b = 0), solving quadratic equations (ax² + bx + c = 0), rearranging formulas to make a specific variable the subject, expanding and factorising brackets, and verifying homework and exam answers. This calculator is used by GCSE and A-Level students across the UK.
Last updated: February 2026 | Verified with latest UK rates
Pro Tips for Accurate Results
- Double-check your input values before calculating
- Use the correct unit format (metric or imperial)
- For complex calculations, break them into smaller steps
- Bookmark this page for quick future access
Understanding Your Results
Our Equation Solver provides:
- Instant calculations - Results appear immediately
- Accurate formulas - Based on official UK standards
- Clear explanations - Understand how results are derived
- 2025/26 updated - Using current rates and regulations
Common Questions
Is this calculator free?
Yes, all our calculators are 100% free to use with no registration required.
Are the results accurate?
Our calculators use verified formulas and are regularly updated for accuracy.
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