Solve x^2+28x+192=0

$${ x }^{ 2 } +28x+192=0$$

Solve for x

$$x=-16$$

$$x=-12$$

Steps Using Factoring

To solve the equation, factor x^{2}+28x+192 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.

$$a+b=28 ab=192$$

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 192.

$$1,192 2,96 3,64 4,48 6,32 8,24 12,16$$

Calculate the sum for each pair.

$$1+192=193 2+96=98 3+64=67 4+48=52 6+32=38 8+24=32 12+16=28$$

The solution is the pair that gives sum 28.

$$a=12 b=16$$

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

$$\left(x+12\right)\left(x+16\right)$$

To find equation solutions, solve x+12=0 and x+16=0.

$$x=-12 x=-16$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+192. To find a and b, set up a system to be solved.

$$a+b=28 ab=1\times 192=192$$

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 192.

$$1,192 2,96 3,64 4,48 6,32 8,24 12,16$$

Calculate the sum for each pair.

$$1+192=193 2+96=98 3+64=67 4+48=52 6+32=38 8+24=32 12+16=28$$

The solution is the pair that gives sum 28.

$$a=12 b=16$$

Rewrite x^{2}+28x+192 as \left(x^{2}+12x\right)+\left(16x+192\right).

$$\left(x^{2}+12x\right)+\left(16x+192\right)$$

Factor out x in the first and 16 in the second group.

$$x\left(x+12\right)+16\left(x+12\right)$$

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

$$\left(x+12\right)\left(x+16\right)$$

To find equation solutions, solve x+12=0 and x+16=0.

$$x=-12 x=-16$$

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.

$$x^{2}+28x+192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 28 for b, and 192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\times 192}}{2}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\times 192}}{2}$$

Multiply -4 times 192.

$$x=\frac{-28±\sqrt{784-768}}{2}$$

Add 784 to -768.

$$x=\frac{-28±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-28±4}{2}$$

Now solve the equation x=\frac{-28±4}{2} when ± is plus. Add -28 to 4.

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

Divide -24 by 2.

$$x=-12$$

Now solve the equation x=\frac{-28±4}{2} when ± is minus. Subtract 4 from -28.

$$x=-\frac{32}{2}$$

Divide -32 by 2.

$$x=-16$$

The equation is now solved.

$$x=-12 x=-16$$

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.

$$x^{2}+28x+192=0$$

Subtract 192 from both sides of the equation.

$$x^{2}+28x+192-192=-192$$

Subtracting 192 from itself leaves 0.

$$x^{2}+28x=-192$$

Divide 28, the coefficient of the x term, by 2 to get 14. Then add the square of 14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+28x+14^{2}=-192+14^{2}$$

Square 14.

$$x^{2}+28x+196=-192+196$$

Add -192 to 196.

$$x^{2}+28x+196=4$$

Factor x^{2}+28x+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+14=2 x+14=-2$$

Subtract 14 from both sides of the equation.

$$x=-12 x=-16$$