Solved on Sep 27, 2023

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

Substitute the values of $a$ and $b$ into the equation to find $d$ .
$d = \frac{8}{2 \cdot1}$

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

Substitute the values of $a$ and $b$ into the equation to find $d$ .
$d = \frac{8}{2 \cdot1}$

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

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

STEP 1

Assumptions1. The equation given is x+8x+8=0x^{}+8 x+8=0x+8x+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}x=a±bc​, where aaa, bbb, and ccc are integers

STEP 2

The general form of a quadratic equation is ax2+bx+c=0ax^{2}+bx+c=0ax2+bx+c=0. Comparing this with the given equation, we get a=1a=1a=1, b=8b=8b=8, and c=8c=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(x+d)2=e. To do this, we need to find the value of ddd which is b/2ab/2ab/2a.d=b2ad = \frac{b}{2a}d=2ab​

STEP 4

Substitute the values of aaa and bbb into the equation to find ddd.d=82⋅1d = \frac{8}{2 \cdot1}d=2⋅18​

STEP 5

Calculate the value of ddd.d=82=4d = \frac{8}{2} =4d=28​=4

STEP 6

Now, rewrite the equation as (x+d)2−d2=−c(x+d)^2 - d^2 = -c(x+d)2−d2=−c.(x+4)2−42=−8(x+4)^2 -4^2 = -8(x+4)2−42=−8

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

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

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

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

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

Substitute the values of $a$ and $b$ into the equation to find $d$ .
$d = \frac{8}{2 \cdot1}$

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

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

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

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

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

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

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

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