Solve x^2/8+1=x+7

$$\frac { x ^ { 2 } } { 8 } + 1 = x + 7$$

Solve for x

$$x=-4$$

$$x=12$$

Steps Using Factoring

Multiply both sides of the equation by 8.

$$x^{2}+8=8x+56$$

Subtract 8x from both sides.

$$x^{2}+8-8x=56$$

Subtract 56 from both sides.

$$x^{2}+8-8x-56=0$$

Subtract 56 from 8 to get -48.

$$x^{2}-48-8x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-8x-48=0$$

To solve the equation, factor x^{2}-8x-48 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-8 ab=-48$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Calculate the sum for each pair.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

The solution is the pair that gives sum -8.

$$a=-12 b=4$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-12\right)\left(x+4\right)$$

To find equation solutions, solve x-12=0 and x+4=0.

$$x=12 x=-4$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 8.

$$x^{2}+8=8x+56$$

Subtract 8x from both sides.

$$x^{2}+8-8x=56$$

Subtract 56 from both sides.

$$x^{2}+8-8x-56=0$$

Subtract 56 from 8 to get -48.

$$x^{2}-48-8x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-8x-48=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-48. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\left(-48\right)=-48$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Calculate the sum for each pair.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

The solution is the pair that gives sum -8.

$$a=-12 b=4$$

Rewrite x^{2}-8x-48 as \left(x^{2}-12x\right)+\left(4x-48\right).

$$\left(x^{2}-12x\right)+\left(4x-48\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(x-12\right)+4\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(x+4\right)$$

To find equation solutions, solve x-12=0 and x+4=0.

$$x=12 x=-4$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 8.

$$x^{2}+8=8x+56$$

Subtract 8x from both sides.

$$x^{2}+8-8x=56$$

Subtract 56 from both sides.

$$x^{2}+8-8x-56=0$$

Subtract 56 from 8 to get -48.

$$x^{2}-48-8x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-8x-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-48\right)}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-48\right)}}{2}$$

Multiply -4 times -48.

$$x=\frac{-\left(-8\right)±\sqrt{64+192}}{2}$$

Add 64 to 192.

$$x=\frac{-\left(-8\right)±\sqrt{256}}{2}$$

Take the square root of 256.

$$x=\frac{-\left(-8\right)±16}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±16}{2}$$

Now solve the equation x=\frac{8±16}{2} when ± is plus. Add 8 to 16.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{8±16}{2} when ± is minus. Subtract 16 from 8.

$$x=-\frac{8}{2}$$

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=12 x=-4$$

Steps for Completing the Square

Multiply both sides of the equation by 8.

$$x^{2}+8=8x+56$$

Subtract 8x from both sides.

$$x^{2}+8-8x=56$$

Subtract 8 from both sides.

$$x^{2}-8x=56-8$$

Subtract 8 from 56 to get 48.

$$x^{2}-8x=48$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=48+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=48+16$$

Add 48 to 16.

$$x^{2}-8x+16=64$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{64}$$

Simplify.

$$x-4=8 x-4=-8$$

Add 4 to both sides of the equation.

$$x=12 x=-4$$