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

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

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((5x^2)/(5)-(40x)/(5)+(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`