Solve x^2+56=-18x

$$x ^ { 2 } + 56 = - 18 x$$

Solve for x

$$x=-14$$

$$x=-4$$

Steps Using Factoring

Add 18x to both sides.

$$x^{2}+56+18x=0$$

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

$$x^{2}+18x+56=0$$

To solve the equation, factor x^{2}+18x+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.

$$a+b=18 ab=56$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 56.

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

Calculate the sum for each pair.

$$1+56=57 2+28=30 4+14=18 7+8=15$$

The solution is the pair that gives sum 18.

$$a=4 b=14$$

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

$$\left(x+4\right)\left(x+14\right)$$

To find equation solutions, solve x+4=0 and x+14=0.

$$x=-4 x=-14$$

Steps Using Factoring By Grouping

Add 18x to both sides.

$$x^{2}+56+18x=0$$

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

$$x^{2}+18x+56=0$$

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.

$$a+b=18 ab=1\times 56=56$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 56.

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

Calculate the sum for each pair.

$$1+56=57 2+28=30 4+14=18 7+8=15$$

The solution is the pair that gives sum 18.

$$a=4 b=14$$

Rewrite x^{2}+18x+56 as \left(x^{2}+4x\right)+\left(14x+56\right).

$$\left(x^{2}+4x\right)+\left(14x+56\right)$$

Factor out x in the first and 14 in the second group.

$$x\left(x+4\right)+14\left(x+4\right)$$

Factor out common term x+4 by using distributive property.

$$\left(x+4\right)\left(x+14\right)$$

To find equation solutions, solve x+4=0 and x+14=0.

$$x=-4 x=-14$$

Steps Using the Quadratic Formula

Add 18x to both sides.

$$x^{2}+56+18x=0$$

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.

$$x^{2}+18x+56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 18 for b, and 56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 56}}{2}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 56}}{2}$$

Multiply -4 times 56.

$$x=\frac{-18±\sqrt{324-224}}{2}$$

Add 324 to -224.

$$x=\frac{-18±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-18±10}{2}$$

Now solve the equation x=\frac{-18±10}{2} when ± is plus. Add -18 to 10.

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

Divide -8 by 2.

$$x=-4$$

Now solve the equation x=\frac{-18±10}{2} when ± is minus. Subtract 10 from -18.

$$x=-\frac{28}{2}$$

Divide -28 by 2.

$$x=-14$$

The equation is now solved.

$$x=-4 x=-14$$

Steps for Completing the Square

Add 18x to both sides.

$$x^{2}+56+18x=0$$

Subtract 56 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+18x=-56$$

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+18x+9^{2}=-56+9^{2}$$

Square 9.

$$x^{2}+18x+81=-56+81$$

Add -56 to 81.

$$x^{2}+18x+81=25$$

Factor x^{2}+18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+9\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+9=5 x+9=-5$$

Subtract 9 from both sides of the equation.

$$x=-4 x=-14$$