Math

Question Solve the quadratic equation x2+8x+8=0x^2 + 8x + 8 = 0 by completing the square. Express the solutions in the form x=a±bcx = a \pm b \sqrt{c}, where bb and cc are integers.

Studdy Solution

STEP 1

Assumptions1. The equation given is x+8x+8=0x^{}+8 x+8=0 . We need to solve this equation by first completing the square3. The solution should be in the form x=a±bcx=a \pm b \sqrt{c}, where aa, bb, and cc are integers

STEP 2

The general form of a quadratic equation is ax2+bx+c=0ax^{2}+bx+c=0. Comparing this with the given equation, we get a=1a=1, b=8b=8, and c=8c=8.

STEP 3

The first step in completing the square is to rewrite the equation in the form (x+d)2=e(x+d)^2 = e. To do this, we need to find the value of dd which is b/2ab/2a.
d=b2ad = \frac{b}{2a}

STEP 4

Substitute the values of aa and bb into the equation to find dd.
d=821d = \frac{8}{2 \cdot1}

STEP 5

Calculate the value of dd.
d=82=4d = \frac{8}{2} =4

STEP 6

Now, rewrite the equation as (x+d)2d2=c(x+d)^2 - d^2 = -c.
(x+4)242=8(x+4)^2 -4^2 = -8

STEP 7

implify the equation.
(x+4)216=(x+4)^2 -16 = -

STEP 8

Rearrange the equation to get (x+d)2(x+d)^2 on one side.
(x+4)2=168(x+4)^2 =16 -8

STEP 9

Calculate the right side of the equation.
(x+4)2=8(x+4)^2 =8

STEP 10

Now, take the square root of both sides of the equation to solve for xx. Remember to consider both the positive and negative roots.
x+4=±8x+4 = \pm \sqrt{8}

STEP 11

Subtract4 from both sides of the equation to isolate xx.
x=4±8x = -4 \pm \sqrt{8}

STEP 12

The number under the square root,8, can be simplified as 222 \sqrt{2}.
x=4±22x = -4 \pm2 \sqrt{2}So the solutions to the equation x2+8x+8=0x^{2}+8 x+8=0 are x=4+22x = -4 +2 \sqrt{2} and x=422x = -4 -2 \sqrt{2}.

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