Solve x^2+8x-513

$${ x }^{ 2 } +8x-513$$

Factor

$$\left(x-19\right)\left(x+27\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-513. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\left(-513\right)=-513$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -513.

$$-1,513 -3,171 -9,57 -19,27$$

Calculate the sum for each pair.

$$-1+513=512 -3+171=168 -9+57=48 -19+27=8$$

The solution is the pair that gives sum 8.

$$a=-19 b=27$$

Rewrite x^{2}+8x-513 as \left(x^{2}-19x\right)+\left(27x-513\right).

$$\left(x^{2}-19x\right)+\left(27x-513\right)$$

Factor out x in the first and 27 in the second group.

$$x\left(x-19\right)+27\left(x-19\right)$$

Factor out common term x-19 by using distributive property.

$$\left(x-19\right)\left(x+27\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$x^{2}+8x-513=0$$

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.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-513\right)}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-513\right)}}{2}$$

Multiply -4 times -513.

$$x=\frac{-8±\sqrt{64+2052}}{2}$$

Add 64 to 2052.

$$x=\frac{-8±\sqrt{2116}}{2}$$

Take the square root of 2116.

$$x=\frac{-8±46}{2}$$

Now solve the equation x=\frac{-8±46}{2} when ± is plus. Add -8 to 46.

$$x=\frac{38}{2}$$

Divide 38 by 2.

$$x=19$$

Now solve the equation x=\frac{-8±46}{2} when ± is minus. Subtract 46 from -8.

$$x=-\frac{54}{2}$$

Divide -54 by 2.

$$x=-27$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 19 for x_{1} and -27 for x_{2}.

$$x^{2}+8x-513=\left(x-19\right)\left(x-\left(-27\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$x^{2}+8x-513=\left(x-19\right)\left(x+27\right)$$

Evaluate

$$\left(x-19\right)\left(x+27\right)$$