How do you solve using the completing the square method x^2+24x+90=0?

2 Answers
Apr 29, 2016

(x+12)^2 -54=0

Explanation:

The expression (x+a)^2 expands as x^2 + 2ax + a^2
so to complete the square we use half the coefficient of the middle term to be a.

We then subtract the equivalent of a^2 (in this case 12^2) and add the final term (90).

(x+12)^2 -144+90
=(x+12)^2 -54

Apr 29, 2016

=>x=-12+-3sqrt(6)" " as exact values

=> x~~-4.65" and "-19.35" " to 2 decimal places

Explanation:

Standard for " "y=ax^2+bx+c"

Write as" "y=a(x+b/(2a))^2 + c + (-b^2/(4a))

The purpose of the b^2/(4a) is to mathematically remove an error that have been introduced by building a(x+b/(2a))^2

If you square b/(2a) then multiply it out by the variable 'a' in front of the bracket you have introduced a value that was not in the original equation. So you remove it by subtraction.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given:" "x^2+24x+90=0" " Note that a=1

Write as:" "y=(x+12)^2+90+k

But k=-(12)^2/4 = -144

color(brown)(" "=>y=(x+12)^2+90+k)color(blue)(" "->" "y=(x+12)^2-54)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

x^2+24x+90=y=0=(x+12)^2-54

So (x+12)^2=+54

=>sqrt((x+12)^2)=sqrt(54)

=>x+12=+-sqrt(6xx3^2)

=>x=-12+-3sqrt(6)

=> x~~-4.65" and "-19.35 to 2 decimal places
Tony BTony B