Solving equations

Maths · Algebra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Solving an equation means finding the value of the unknown that makes both sides equal. For a linear equation, do the same thing to both sides until x stands alone: expand brackets, clear fractions, collect the x terms on one side and the numbers on the other.

A quadratic equation has an x² term and can have two solutions. First make one side zero. Then factorise, complete the square or use the quadratic formula.

Some equations cannot be solved exactly. A graph gives approximate solutions where two graphs cross. Iteration gives them by putting each answer back into a formula again and again until the values settle.

Rules to remember

Linear: whatever you do to one side, do to the other side too.
Fractions: multiply every term by the common denominator first.
Factorising: if (x − p)(x − q) = 0 then x = p or x = q.
Quadratic formula for ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ (2a)
Completing the square: x² + bx = (x + b/2)² − (b/2)²
If b² − 4ac is negative there are no real solutions; if it is zero there is one repeated solution.
Iteration: put x₀ into the formula to get x₁, put x₁ in to get x₂, and so on.

Common mistake

Students factorise a quadratic while the other side is not zero, or give only one solution. Always rearrange to "= 0" first and write both values of x.

Worked example

Solve 3x² − 10x − 8 = 0.

Find two numbers that multiply to 3 × (−8) = −24 and add to −10: these are −12 and 2.
Split the middle term: 3x² − 12x + 2x − 8 = 3x(x − 4) + 2(x − 4) = (3x + 2)(x − 4).
Set each bracket to zero: 3x + 2 = 0 gives x = −2/3, and x − 4 = 0 gives x = 4.

Answer: x = −2/3 or x = 4

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