Solve 1/x*16=1+28/x^2

$$\frac{ 1 }{ x } \times 16=1+ \frac{ 28 }{ { x }^{ 2 } }$$

Solve for x

$$x=2$$

$$x=14$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}, the least common multiple of x,x^{2}.

$$x\times 1\times 16=x^{2}+28$$

Multiply 1 and 16 to get 16.

$$x\times 16=x^{2}+28$$

Subtract x^{2} from both sides.

$$x\times 16-x^{2}=28$$

Subtract 28 from both sides.

$$x\times 16-x^{2}-28=0$$

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

$$-x^{2}+16x-28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-28. To find a and b, set up a system to be solved.

$$a+b=16 ab=-\left(-28\right)=28$$

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 28.

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

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 16.

$$a=14 b=2$$

Rewrite -x^{2}+16x-28 as \left(-x^{2}+14x\right)+\left(2x-28\right).

$$\left(-x^{2}+14x\right)+\left(2x-28\right)$$

Factor out -x in the first and 2 in the second group.

$$-x\left(x-14\right)+2\left(x-14\right)$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(-x+2\right)$$

To find equation solutions, solve x-14=0 and -x+2=0.

$$x=14 x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}, the least common multiple of x,x^{2}.

$$x\times 1\times 16=x^{2}+28$$

Multiply 1 and 16 to get 16.

$$x\times 16=x^{2}+28$$

Subtract x^{2} from both sides.

$$x\times 16-x^{2}=28$$

Subtract 28 from both sides.

$$x\times 16-x^{2}-28=0$$

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.

$$-x^{2}+16x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 16 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-16±\sqrt{256+4\left(-28\right)}}{2\left(-1\right)}$$

Multiply 4 times -28.

$$x=\frac{-16±\sqrt{256-112}}{2\left(-1\right)}$$

Add 256 to -112.

$$x=\frac{-16±\sqrt{144}}{2\left(-1\right)}$$

Take the square root of 144.

$$x=\frac{-16±12}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-16±12}{-2}$$

Now solve the equation x=\frac{-16±12}{-2} when ± is plus. Add -16 to 12.

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

Divide -4 by -2.

$$x=2$$

Now solve the equation x=\frac{-16±12}{-2} when ± is minus. Subtract 12 from -16.

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

Divide -28 by -2.

$$x=14$$

The equation is now solved.

$$x=2 x=14$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}, the least common multiple of x,x^{2}.

$$x\times 1\times 16=x^{2}+28$$

Multiply 1 and 16 to get 16.

$$x\times 16=x^{2}+28$$

Subtract x^{2} from both sides.

$$x\times 16-x^{2}=28$$

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.

$$-x^{2}+16x=28$$

Divide both sides by -1.

$$\frac{-x^{2}+16x}{-1}=\frac{28}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{16}{-1}x=\frac{28}{-1}$$

Divide 16 by -1.

$$x^{2}-16x=\frac{28}{-1}$$

Divide 28 by -1.

$$x^{2}-16x=-28$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-28+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-28+64$$

Add -28 to 64.

$$x^{2}-16x+64=36$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{36}$$

Simplify.

$$x-8=6 x-8=-6$$

Add 8 to both sides of the equation.

$$x=14 x=2$$