Solve 3/2x+7=x-2

$$\frac { 3 } { 2 x } + 7 = x - 2$$

Solve for x

$$x = \frac{\sqrt{87} + 9}{2} \approx 9.163689527$$

$$x=\frac{9-\sqrt{87}}{2}\approx -0.163689527$$

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 2x.

$$3+2x\times 7=2xx+2x\left(-2\right)$$

Multiply 2 and 7 to get 14.

$$3+14x=2xx+2x\left(-2\right)$$

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

$$3+14x=2x^{2}+2x\left(-2\right)$$

Multiply 2 and -2 to get -4.

$$3+14x=2x^{2}-4x$$

Subtract 2x^{2} from both sides.

$$3+14x-2x^{2}=-4x$$

Add 4x to both sides.

$$3+14x-2x^{2}+4x=0$$

Combine 14x and 4x to get 18x.

$$3+18x-2x^{2}=0$$

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.

$$-2x^{2}+18x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 18 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-2\right)\times 3}}{2\left(-2\right)}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-2\right)\times 3}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-18±\sqrt{324+8\times 3}}{2\left(-2\right)}$$

Multiply 8 times 3.

$$x=\frac{-18±\sqrt{324+24}}{2\left(-2\right)}$$

Add 324 to 24.

$$x=\frac{-18±\sqrt{348}}{2\left(-2\right)}$$

Take the square root of 348.

$$x=\frac{-18±2\sqrt{87}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-18±2\sqrt{87}}{-4}$$

Now solve the equation x=\frac{-18±2\sqrt{87}}{-4} when ± is plus. Add -18 to 2\sqrt{87}.

$$x=\frac{2\sqrt{87}-18}{-4}$$

Divide -18+2\sqrt{87} by -4.

$$x=\frac{9-\sqrt{87}}{2}$$

Now solve the equation x=\frac{-18±2\sqrt{87}}{-4} when ± is minus. Subtract 2\sqrt{87} from -18.

$$x=\frac{-2\sqrt{87}-18}{-4}$$

Divide -18-2\sqrt{87} by -4.

$$x=\frac{\sqrt{87}+9}{2}$$

The equation is now solved.

$$x=\frac{9-\sqrt{87}}{2} x=\frac{\sqrt{87}+9}{2}$$

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 2x.

$$3+2x\times 7=2xx+2x\left(-2\right)$$

Multiply 2 and 7 to get 14.

$$3+14x=2xx+2x\left(-2\right)$$

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

$$3+14x=2x^{2}+2x\left(-2\right)$$

Multiply 2 and -2 to get -4.

$$3+14x=2x^{2}-4x$$

Subtract 2x^{2} from both sides.

$$3+14x-2x^{2}=-4x$$

Add 4x to both sides.

$$3+14x-2x^{2}+4x=0$$

Combine 14x and 4x to get 18x.

$$3+18x-2x^{2}=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$18x-2x^{2}=-3$$

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.

$$-2x^{2}+18x=-3$$

Divide both sides by -2.

$$\frac{-2x^{2}+18x}{-2}=-\frac{3}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{18}{-2}x=-\frac{3}{-2}$$

Divide 18 by -2.

$$x^{2}-9x=-\frac{3}{-2}$$

Divide -3 by -2.

$$x^{2}-9x=\frac{3}{2}$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=\frac{3}{2}+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=\frac{3}{2}+\frac{81}{4}$$

Add \frac{3}{2} to \frac{81}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-9x+\frac{81}{4}=\frac{87}{4}$$

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{87}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{87}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{87}}{2} x-\frac{9}{2}=-\frac{\sqrt{87}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{\sqrt{87}+9}{2} x=\frac{9-\sqrt{87}}{2}$$