Remeber the formula: (a - b)² = a² - 2ab + b². Notice that y = x² - 23x + 5, so we want to write this using (x - 3)² = x² - 23x* + 9. Taking 4 from both sides gives: (x - 3)²- 4 = x² - 6x + 5 = y.

We need some simple facts about graphs: (1) y = x²is a parabola (U shaped); (2) if we replace x wih x - 3 we move the graph to the right by 3; (3) if we add -4 to y, the graph moves down by 4.

To find minima: notice that (x - 3)² is always positive or 0, so (x - 3)²+ -4 >= -4. If x is not 3, then (x - 3)² > 0, so y > -4; but if x = 3, we have y = -4, so -4 is the smallest value of y (i.e. a minimum) at (3, -4).

To find roots, we can solve the quadratic y = 0:

(x - 3)² - 4 = 0 <=> (x - 3)² = 4 <=> x - 3 = 2 or x - 3 = -2 <=> x = 5 or x = 1.

(Rememer that x² = a² has two solution: x = a and x = -a.) With this it should be easy to sketch the curve!

Expand (2x+4)(x-3)

Related Maths GCSE answers

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