Solve xx+90x+176=0

$$xx+90x+176=0$$

Solve for x

$$x=-88$$

$$x=-2$$

Steps Using Factoring

Multiply x and x to get x^{2}.

$$x^{2}+90x+176=0$$

To solve the equation, factor x^{2}+90x+176 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.

$$a+b=90 ab=176$$

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 176.

$$1,176 2,88 4,44 8,22 11,16$$

Calculate the sum for each pair.

$$1+176=177 2+88=90 4+44=48 8+22=30 11+16=27$$

The solution is the pair that gives sum 90.

$$a=2 b=88$$

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

$$\left(x+2\right)\left(x+88\right)$$

To find equation solutions, solve x+2=0 and x+88=0.

$$x=-2 x=-88$$

Steps Using Factoring By Grouping

Multiply x and x to get x^{2}.

$$x^{2}+90x+176=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+176. To find a and b, set up a system to be solved.

$$a+b=90 ab=1\times 176=176$$

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 176.

$$1,176 2,88 4,44 8,22 11,16$$

Calculate the sum for each pair.

$$1+176=177 2+88=90 4+44=48 8+22=30 11+16=27$$

The solution is the pair that gives sum 90.

$$a=2 b=88$$

Rewrite x^{2}+90x+176 as \left(x^{2}+2x\right)+\left(88x+176\right).

$$\left(x^{2}+2x\right)+\left(88x+176\right)$$

Factor out x in the first and 88 in the second group.

$$x\left(x+2\right)+88\left(x+2\right)$$

Factor out common term x+2 by using distributive property.

$$\left(x+2\right)\left(x+88\right)$$

To find equation solutions, solve x+2=0 and x+88=0.

$$x=-2 x=-88$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$x^{2}+90x+176=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 90 for b, and 176 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-90±\sqrt{90^{2}-4\times 176}}{2}$$

Square 90.

$$x=\frac{-90±\sqrt{8100-4\times 176}}{2}$$

Multiply -4 times 176.

$$x=\frac{-90±\sqrt{8100-704}}{2}$$

Add 8100 to -704.

$$x=\frac{-90±\sqrt{7396}}{2}$$

Take the square root of 7396.

$$x=\frac{-90±86}{2}$$

Now solve the equation x=\frac{-90±86}{2} when ± is plus. Add -90 to 86.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

Now solve the equation x=\frac{-90±86}{2} when ± is minus. Subtract 86 from -90.

$$x=-\frac{176}{2}$$

Divide -176 by 2.

$$x=-88$$

The equation is now solved.

$$x=-2 x=-88$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$x^{2}+90x+176=0$$

Subtract 176 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+90x=-176$$

Divide 90, the coefficient of the x term, by 2 to get 45. Then add the square of 45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+90x+45^{2}=-176+45^{2}$$

Square 45.

$$x^{2}+90x+2025=-176+2025$$

Add -176 to 2025.

$$x^{2}+90x+2025=1849$$

Factor x^{2}+90x+2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+45\right)^{2}=1849$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+45\right)^{2}}=\sqrt{1849}$$

Simplify.

$$x+45=43 x+45=-43$$

Subtract 45 from both sides of the equation.

$$x=-2 x=-88$$