Equations
Solving equations
- Solve a linear equation by doing the same to both sides until the unknown is alone.
- (Extended) Clear a fractional equation by multiplying out the denominator.
- Simultaneous linear equations: add or subtract to remove one letter.
- $2x + y = 7$ and $3x - y = 8$: adding gives $5x = 15$, so $x = 3$, then $y = 1$.
Practice
Solve 2x + y = 7 and 3x − y = 8. What is x?
Adding the equations: 5x = 15, so x = 3 (then y = 1).
Quadratic equations (Extended)
- By factorising: $x^2 + 5x + 6 = 0 \to (x+2)(x+3) = 0 \to x = -2$ or $-3$.
- By the formula (given in the exam): $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Practice
Solve x² + 5x + 6 = 0 by factorising.
(x + 2)(x + 3) = 0, so x = −2 or x = −3.
Practice
The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a).
This formula solves any quadratic ax² + bx + c = 0.
You've got it
Key idea
- linear: same operation to both sides; fractional: clear the denominator
- simultaneous linear: add/subtract to eliminate a letter
- quadratics: factorise, complete the square, or use the formula