⚖️ Equations

Keep both sides balanced to solve for \(x\)

Difficulty

Click "New Equation" to begin

Solve for \(x\). You may enter a whole number or a fraction like 3/2.

x =

Score: 0/0

The Balance Rule

An equation is like a balance scale: the left side always equals the right side. Whatever you do to one side, you must do to the other side too — then it stays balanced.

Inverse Operations

+ undoes − × undoes ÷

To move a number, do the opposite operation to both sides.

One-Step

\(x + 3 = 7\) \(x = 7 - 3 = 4\)

Subtract 3 from both sides.

Two-Step

\(2x + 3 = 11\) \(2x = 8 \;\Rightarrow\; x = 4\)

Undo the \(+3\) first, then the \(\times 2\).

Variables on Both Sides

\(3x + 2 = x + 10\) \(2x = 8 \;\Rightarrow\; x = 4\)

Collect the \(x\) terms on one side first.

Worked Example

Solve \(3x + 2 = x + 10\)

1. Start: \(3x + 2 = x + 10\)

2. Subtract \(x\) from both sides: \(2x + 2 = 10\)

3. Subtract \(2\) from both sides: \(2x = 8\)

4. Divide both sides by \(2\): \(x = 4\)

5. Check: \(3(4) + 2 = 14\) and \(4 + 10 = 14\) ✓

Tips

Do the same thing to both sides every time.
Undo addition/subtraction first, then multiplication/division.
Collect all the \(x\) terms on one side and the numbers on the other.
Always check by putting your answer back into the original equation.

📐 Expanding & Factoring

“Expanding” means multiplying brackets out. “Factoring” is the reverse — writing an expression as a product of brackets. These identities are worth memorising:

Distributive Law

\(a(b + c) = ab + ac\)

Multiply the outside term into each inside term.

Two Binomials

\((x + a)(x + b)\) \(= x^2 + (a + b)x + ab\)

Add for the middle term, multiply for the last.

Perfect Square

\((a + b)^2 = a^2 + 2ab + b^2\) \((a - b)^2 = a^2 - 2ab + b^2\)

Difference of Squares

\((a + b)(a - b) = a^2 - b^2\)

Perfect Cube

\((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)

Sum & Difference of Cubes

\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

Worked Examples

Expand \((x + 2)(x + 5)\):

1. \((x + 2)(x + 5) = x^2 + (2 + 5)x + (2)(5)\)

2. \(= x^2 + 7x + 10\)

Factor \(x^2 + 7x + 10\):

1. Find two numbers that multiply to \(10\) and add to \(7\): \(2\) and \(5\).

2. \(x^2 + 7x + 10 = (x + 2)(x + 5)\)

Mode

Difficulty

Click "New Problem" to begin

Multiply the brackets out. Fill in the missing coefficient and constant (they may be negative).

x² + x +

Write it as two brackets. (The numbers may be negative — order doesn't matter.)

(x + )(x + )

Expand the expression. Fill in each missing coefficient (they may be negative; enter 0 if a term disappears).

Score: 0/0