Solve x^2+54x=-504

$$x ^ { 2 } + 54 x = - 504$$

Solve for x

$$x=-42$$

$$x=-12$$

Steps Using Factoring

Add 504 to both sides.

$$x^{2}+54x+504=0$$

To solve the equation, factor x^{2}+54x+504 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.

$$a+b=54 ab=504$$

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 504.

$$1,504 2,252 3,168 4,126 6,84 7,72 8,63 9,56 12,42 14,36 18,28 21,24$$

Calculate the sum for each pair.

$$1+504=505 2+252=254 3+168=171 4+126=130 6+84=90 7+72=79 8+63=71 9+56=65 12+42=54 14+36=50 18+28=46 21+24=45$$

The solution is the pair that gives sum 54.

$$a=12 b=42$$

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

$$\left(x+12\right)\left(x+42\right)$$

To find equation solutions, solve x+12=0 and x+42=0.

$$x=-12 x=-42$$

Steps Using Factoring By Grouping

Add 504 to both sides.

$$x^{2}+54x+504=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+504. To find a and b, set up a system to be solved.

$$a+b=54 ab=1\times 504=504$$

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 504.

$$1,504 2,252 3,168 4,126 6,84 7,72 8,63 9,56 12,42 14,36 18,28 21,24$$

Calculate the sum for each pair.

$$1+504=505 2+252=254 3+168=171 4+126=130 6+84=90 7+72=79 8+63=71 9+56=65 12+42=54 14+36=50 18+28=46 21+24=45$$

The solution is the pair that gives sum 54.

$$a=12 b=42$$

Rewrite x^{2}+54x+504 as \left(x^{2}+12x\right)+\left(42x+504\right).

$$\left(x^{2}+12x\right)+\left(42x+504\right)$$

Factor out x in the first and 42 in the second group.

$$x\left(x+12\right)+42\left(x+12\right)$$

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

$$\left(x+12\right)\left(x+42\right)$$

To find equation solutions, solve x+12=0 and x+42=0.

$$x=-12 x=-42$$

Steps Using the Quadratic Formula

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.

$$x^{2}+54x=-504$$

Add 504 to both sides of the equation.

$$x^{2}+54x-\left(-504\right)=-504-\left(-504\right)$$

Subtracting -504 from itself leaves 0.

$$x^{2}+54x-\left(-504\right)=0$$

Subtract -504 from 0.

$$x^{2}+54x+504=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 54 for b, and 504 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-54±\sqrt{54^{2}-4\times 504}}{2}$$

Square 54.

$$x=\frac{-54±\sqrt{2916-4\times 504}}{2}$$

Multiply -4 times 504.

$$x=\frac{-54±\sqrt{2916-2016}}{2}$$

Add 2916 to -2016.

$$x=\frac{-54±\sqrt{900}}{2}$$

Take the square root of 900.

$$x=\frac{-54±30}{2}$$

Now solve the equation x=\frac{-54±30}{2} when ± is plus. Add -54 to 30.

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

Divide -24 by 2.

$$x=-12$$

Now solve the equation x=\frac{-54±30}{2} when ± is minus. Subtract 30 from -54.

$$x=-\frac{84}{2}$$

Divide -84 by 2.

$$x=-42$$

The equation is now solved.

$$x=-12 x=-42$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+54x=-504$$

Divide 54, the coefficient of the x term, by 2 to get 27. Then add the square of 27 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+54x+27^{2}=-504+27^{2}$$

Square 27.

$$x^{2}+54x+729=-504+729$$

Add -504 to 729.

$$x^{2}+54x+729=225$$

Factor x^{2}+54x+729. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+27\right)^{2}=225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+27\right)^{2}}=\sqrt{225}$$

Simplify.

$$x+27=15 x+27=-15$$

Subtract 27 from both sides of the equation.

$$x=-12 x=-42$$