Question

How do you solve `x^2-4.7x=-2.8` by completing the square?

Answer 1

`color(red)(x=4` or `color(red)(x=0.7`

Explanation:

Completing the square:

`:.x^2-4.7x+2.8=0`

`:.x^2-4.7x+(-4.7/2)^2=-2.8+(-4.7/2)^2`

`:.x^2-4.7x+5.5225=-2.8+5.5225`

`:.(x-2.35)^2=2.7225`

`:.sqrt((x-2.35)^2)=sqrt(2.7225)`

`:.x-2.35=+-1.65`

`:.x=2.35+-1.65`

`:.color(red)(x=2.35+1.65=4`

or `:.color(red)(x=2.35-1.65=0.7`

check:

`:.x=(-b+-sqrt(b^2-4ac))/(2a)`

`:.x=(-(-4.7)+-sqrt((-4.7)^2-4(1)(2.8)))/(2a)`

`:.x=(-(-4.7)+-sqrt(22.09-11.2))/(2a)`

`:.x=(4.7+-sqrt10.89)/2`

`:.x=(4.7+-3.3)/2`

`:.x=(4.7+3.3)/2`

`:.color(red)(x=8/2=4`

or `:. x=(4.7-3.3)/2`

`:.color(red)(x=1.4/2=0.7`

Answer 2

`x=4" "` or `" "x=0.7`

Explanation:

The difference of squares identity can be written:

`a^2-b^2 = (a-b)(a+b)`

Use this with `a=(20x-47)` and `b=33` below.

Given:

`x^2-4.7x=-2.8`

Add `2.8` to both sides to get:

`x^2-4.7x+2.8 = 0`

Multiply through by `400` to allow us to complete the square using integers...

`0 = 400(x^2-4.7x+2.8)`

`color(white)(0) = 400x^2-1880x+1120`

`color(white)(0) = (20x)^2-2(20x)(47)+47^2-1089`

`color(white)(0) = (20x-47)^2-33^2`

`color(white)(0) = ((20x-47)-33)((20x-47)+33)`

`color(white)(0) = (20x-80)(20x-14)`

`color(white)(0) = 40(x-4)(10x-7)`

So:

`x=4" "` or `" "x = 7/10 = 0.7`