How do you solve x^2 - 8x - 20 = 0x2−8x−20=0 by completing the square?

2 Answers
George C.
May 14, 2016

x=10x=10 or x=-2x=−2

Explanation:

In addition to completing the square we can use the difference of squares identity:

a^2-b^2 = (a-b)(a+b)a2−b2=(a−b)(a+b)

with a=(x-4)a=(x−4) and b=6b=6 as follows:

0 = x^2-8x-200=x2−8x−20

=(x-4)^2-4^2-20=(x−4)2−42−20

=(x-4)^2-16-20=(x−4)2−16−20

=(x-4)^2-36=(x−4)2−36

=(x-4)^2-6^2=(x−4)2−62

=((x-4)-6)((x-4)+6)=((x−4)−6)((x−4)+6)

=(x-10)(x+2)=(x−10)(x+2)

So x=10x=10 or x=-2x=−2

Sheffy
May 14, 2016

x=10, -2x=10,−2

Explanation:

color(red)("We know that " (a+b)^2=a^2 +2ab+b^2We know that (a+b)2=a2+2ab+b2

color(red)("And "(a-b)^2=a^2 -2ab+b^2And (a−b)2=a2−2ab+b2

x^2-8x-20=0x2−8x−20=0

This can be written as x^2+(2xx x xx-4)-20=0x2+(2×x×−4)−20=0

Let a=xa=x and b=-4b=−4

color(red)("Now, " (x-4)^2=x^2 -8x+16Now, (x−4)2=x2−8x+16

x^2-8x-20=0x2−8x−20=0

x^2-8x=20x2−8x=20

Add 16 to both sides:

x^2-8xcolor(red)(+16)=20color(red)(+16)x2−8x+16=20+16

(x-4)^2=36(x−4)2=36

We know that 36=6^2 " and " 36=-6^2 36=62 and 36=−62

Find the square root of both the sides:

x-4=+-6x−4=±6

When, x-4=6x−4=6

Add 4 to both sides:

x-4color(red)(+4)=6color(red)(+4)x−4+4=6+4

x=10x=10

When, x-4=-6x−4=−6

Add 4 to both sides:

x-4color(red)(+4)=-6color(red)(+4)x−4+4=−6+4

x=-2x=−2

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