Evaluate the expression (2x + 5)^2 - 4(x + 3)(x - 3) at x = 1/20.

Step-by-step solution

1. 1

   Simplify before substituting. Plugging $x=\tfrac1{20}$ in straight away forces you to square a fraction and multiply three brackets. Expanding first is both faster and less error-prone — and it reveals that the quadratic terms cancel.

2. 2

   Expand the square.

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

3. 3

   Expand the product as a difference of squares. $(x+3)(x-3)$ is the classic $a^2-b^2$ pattern:

   $(x+3)(x-3)=x^2-9\quad\Longrightarrow\quad 4(x+3)(x-3)=4x^2-36$

4. 4

   Subtract and watch the second minus sign.

   $(4x^2+20x+25)-(4x^2-36)=4x^2+20x+25-4x^2+36=20x+61$

   The $4x^2$ terms cancel and $-(-36)$ becomes $+36$; the whole expression is linear.

5. 5

   Substitute the fraction into the linear form. Now $x=\tfrac1{20}$ is easy, because $20$ and $\tfrac1{20}$ are reciprocals:

   $20\cdot\frac{1}{20}+61=1+61=62$

6. 6

   Check with the unsimplified expression. With $x=0.05$: $(2(0.05)+5)^2=5.1^2=26.01$ and $4(3.05)(-2.95)=-35.99$, so $26.01-(-35.99)=26.01+35.99=62$. $\checkmark$