Solve x+2/x+x=74/x

$$\frac{ x+2 }{ x } +x= \frac{ 74 }{ x }$$

Solve for x

$$x=-9$$

$$x=8$$

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.

$$x+2+xx=74$$

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

$$x+2+x^{2}=74$$

Subtract 74 from both sides.

$$x+2+x^{2}-74=0$$

Subtract 74 from 2 to get -72.

$$x-72+x^{2}=0$$

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.

$$x^{2}+x-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-72\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-72\right)}}{2}$$

Multiply -4 times -72.

$$x=\frac{-1±\sqrt{1+288}}{2}$$

Add 1 to 288.

$$x=\frac{-1±\sqrt{289}}{2}$$

Take the square root of 289.

$$x=\frac{-1±17}{2}$$

Now solve the equation x=\frac{-1±17}{2} when ± is plus. Add -1 to 17.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{-1±17}{2} when ± is minus. Subtract 17 from -1.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=8 x=-9$$

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.

$$x+2+xx=74$$

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

$$x+2+x^{2}=74$$

Subtract 2 from both sides.

$$x+x^{2}=74-2$$

Subtract 2 from 74 to get 72.

$$x+x^{2}=72$$

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.

$$x^{2}+x=72$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=72+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=72+\frac{1}{4}$$

Add 72 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{289}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{289}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{289}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{17}{2} x+\frac{1}{2}=-\frac{17}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=8 x=-9$$