Solve x^2-56=x

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

Solve for x

$$x=-7$$

$$x=8$$

Steps Using Factoring

Subtract x from both sides.

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

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

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

To solve the equation, factor x^{2}-x-56 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.

$$a+b=-1 ab=-56$$

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 -56.

$$1,-56 2,-28 4,-14 7,-8$$

Calculate the sum for each pair.

$$1-56=-55 2-28=-26 4-14=-10 7-8=-1$$

The solution is the pair that gives sum -1.

$$a=-8 b=7$$

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

$$\left(x-8\right)\left(x+7\right)$$

To find equation solutions, solve x-8=0 and x+7=0.

$$x=8 x=-7$$

Steps Using Factoring By Grouping

Subtract x from both sides.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-56. To find a and b, set up a system to be solved.

$$a+b=-1 ab=1\left(-56\right)=-56$$

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 -56.

$$1,-56 2,-28 4,-14 7,-8$$

Calculate the sum for each pair.

$$1-56=-55 2-28=-26 4-14=-10 7-8=-1$$

The solution is the pair that gives sum -1.

$$a=-8 b=7$$

Rewrite x^{2}-x-56 as \left(x^{2}-8x\right)+\left(7x-56\right).

$$\left(x^{2}-8x\right)+\left(7x-56\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(x-8\right)+7\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x+7\right)$$

To find equation solutions, solve x-8=0 and x+7=0.

$$x=8 x=-7$$

Steps Using the Quadratic Formula

Subtract x from both sides.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-56\right)}}{2}$$

Multiply -4 times -56.

$$x=\frac{-\left(-1\right)±\sqrt{1+224}}{2}$$

Add 1 to 224.

$$x=\frac{-\left(-1\right)±\sqrt{225}}{2}$$

Take the square root of 225.

$$x=\frac{-\left(-1\right)±15}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±15}{2}$$

Now solve the equation x=\frac{1±15}{2} when ± is plus. Add 1 to 15.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{1±15}{2} when ± is minus. Subtract 15 from 1.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

The equation is now solved.

$$x=8 x=-7$$

Steps for Completing the Square

Subtract x from both sides.

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

Add 56 to both sides. Anything plus zero gives itself.

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

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=56+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=56+\frac{1}{4}$$

Add 56 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{225}{4}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{15}{2} x-\frac{1}{2}=-\frac{15}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=8 x=-7$$