Algebraic proof (AQA GCSE Further Maths): Revision Notes

Worked Example: Proving an Expression is a Square Number

Let's examine how to prove that $2a^3 - a^2(2a - 9)$ is a square number when $a$ is an integer:

Step 1: Expand the expression Start by expanding the bracketed terms: $2a^3 - a^2(2a - 9) = 2a^3 - 2a^3 + 9a^2$

Step 2: Simplify Combine like terms: $= 9a^2$

Step 3: Recognise the perfect square Factor out the perfect square: $= (3a)^2$

Step 4: State the conclusion Since $a$ is an integer, $3a$ is also an integer. Therefore, $2a^3 - a^2(2a - 9)$ is a square number when $a$ is an integer.

Worked Example: Proving an Expression is Always Positive

Here's how to prove that $x^2 - 8x + 18$ is always positive:

Step 1: Complete the square To complete the square for $x^2 - 8x + 18$:

• Take half the coefficient of $x$: $-8 ÷ 2 = -4$
• Square this value: $(-4)^2 = 16$
• Rewrite: $x^2 - 8x + 18 = (x - 4)^2 - 16 + 18 = (x - 4)^2 + 2$

Step 2: Use properties of squares Since $(x - 4)^2 ≥ 0$ for all real values of $x$ (because any number squared is never negative), we know: $(x - 4)^2 + 2 ≥ 0 + 2 = 2$

Step 3: State the conclusion Therefore, $x^2 - 8x + 18 > 0$ for all real values of $x$, proving it's always positive.

Worked Example: Proving Inequalities with Rational Functions

For a function like $f(x) = \frac{2c + cx}{2d + dx}$ where $c > d$ and both are positive:

Step 1: Factor numerator and denominator $f(x) = \frac{c(2 + x)}{d(2 + x)}$

Step 2: Cancel common factors $f(x) = \frac{c}{d}$

Step 3: Apply given conditions Since $c > d$ and $d$ is positive, we know $\frac{c}{d} > 1$

Step 4: State the conclusion Therefore, $f(x) > 1$