Year 10 Quadratic expressions and equations: revision mind map

1Expanding expressions

Multiply every term in one bracket by every term in the other.

$(x + 3)(x - 7) = x^2 - 4x - 21$

$(2x - 5)^2 = 4x^2 - 20x + 25$

$(3x - 1)(3x + 1) = 9x^2 - 1$

2Factorising expressions

Take out the HCF first, then look for the difference of two squares.

$12x^2 - 8x = 4x(3x - 2)$

$25x^2 - 16 = (5x - 4)(5x + 4)$

$x^2 - 5 = (x - \sqrt5)(x + \sqrt5)$

3Algebraic fractions: × and ÷

Factorise everything, cancel common factors, then multiply. To divide, flip the second fraction.

$\dfrac{x^2 - 9}{x + 3} \times \dfrac{2}{x - 3} = 2$

$\dfrac{x^2 + 2x}{4} \div \dfrac{x}{8} = 2(x + 2)$

4Monic trinomials

$x^2 + bx + c$: two numbers that multiply to $c$ and add to $b$.

$x^2 - 2x - 15 = (x - 5)(x + 3)$

$x^2 + 9x + 20 = (x + 4)(x + 5)$

5Non-monic trinomials

Find two numbers that multiply to $ac$ and add to $b$, split the middle term, then group.

$3x^2 + 10x + 8$: $ac = 24$, use 6 and 4

$3x(x + 2) + 4(x + 2) = (3x + 4)(x + 2)$

6Completing the square

Add and subtract $\left(\tfrac b2\right)^2$: $x^2 + bx = \left(x + \tfrac b2\right)^2 - \left(\tfrac b2\right)^2$

$x^2 + 6x + 4 = (x + 3)^2 - 5$

$= (x + 3 - \sqrt5)(x + 3 + \sqrt5)$

7Solving by factorising

Make one side zero, factorise, then use the Null Factor Law.

$2x^2 - 5x - 3 = 0 \Rightarrow (2x + 1)(x - 3) = 0$

$x = -\tfrac12$ or $x = 3$

8Applications of quadratics

Write the equation, solve it, then check which answers make sense.

A ball's height is $h = 20t - 5t^2$. When is it 15 m up?

$5t^2 - 20t + 15 = 0 \Rightarrow t^2 - 4t + 3 = 0$, so $t = 1$ or $3$ s

Two answers: once going up, once coming down.

9Solving by completing the square

Complete the square, then take the square root of both sides (±).

$x^2 - 4x - 1 = 0 \Rightarrow (x - 2)^2 = 5$

$x = 2 \pm \sqrt5$

10The quadratic formula

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The discriminant $\Delta = b^2 - 4ac$ gives the number of solutions.

$2x^2 + 3x - 1 = 0$: $\Delta = 9 + 8 = 17$, $x = \dfrac{-3 \pm \sqrt{17}}{4}$

$\Delta > 0$: two. $\Delta = 0$: one. $\Delta < 0$: none.