Question

Cedric completed the square for the quadratic expression 5x^2−40x+15 in order to determine the minimum value of the expression, as shown.
Step 1: 5(x^2−8x+3)
Step 2: 5(x^2−8x+16−16+3)
Step 3: 5((x+4)^2−16+3)
Step 4: 5((x+4)^2−13)
Step 5: 5(x+4)^2−65
What mistake, if any, did Cedric make?

Answer 1

Answer:

Step 3 is not correct (should be $(x-4)^2$, not $(x+4)^2$

Step-by-step explanation:

The expression that we have to simplify is:

$5x^2-40x+15$

We proceed step-by-step, in order to find the mistake did by Cedric:

1) First, we factorize the 5 outside:

$5(\frac{5x^2}{5}-\frac{40x}{5}+\frac{15}{5})=5(x^2-40x+3)$ --> this step is correct

2) We add +16 and -16 inside the brackets:

$5(x^2-8x+16-16+3)$ --> this step is correct

3) We rewrite the term $(x^2-8x+16)$ as the square of a bynomial, which is $(x-4)^2$, so the expression becomes

$5((x-4)^2-16+3)$ --> we notice that this step is wrong: in fact, Cedric wrote $(x+4)^2$, which is not correct.

4) Now we continue: we rewrite -16+3 as -13,

$5((x-4)^2-13)$

5) Finally, we multiply the 5 by the terms in the bracket:

$=5(x-4)^2-65$