Solve x+3=9/x+1/2

$$x + 3 = 9 / x + 1 / 2$$

Solve for x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

$$x=2$$

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 2x, the least common multiple of x,2.

$$2xx+2x\times 3=2\times 9+2x\times \frac{1}{2}$$

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

$$2x^{2}+2x\times 3=2\times 9+2x\times \frac{1}{2}$$

Multiply 2 and 3 to get 6.

$$2x^{2}+6x=2\times 9+2x\times \frac{1}{2}$$

Multiply 2 and 9 to get 18.

$$2x^{2}+6x=18+2x\times \frac{1}{2}$$

Cancel out 2 and 2.

$$2x^{2}+6x=18+x$$

Subtract 18 from both sides.

$$2x^{2}+6x-18=x$$

Subtract x from both sides.

$$2x^{2}+6x-18-x=0$$

Combine 6x and -x to get 5x.

$$2x^{2}+5x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 5 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\left(-18\right)}}{2\times 2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 2\left(-18\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-5±\sqrt{25-8\left(-18\right)}}{2\times 2}$$

Multiply -8 times -18.

$$x=\frac{-5±\sqrt{25+144}}{2\times 2}$$

Add 25 to 144.

$$x=\frac{-5±\sqrt{169}}{2\times 2}$$

Take the square root of 169.

$$x=\frac{-5±13}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-5±13}{4}$$

Now solve the equation x=\frac{-5±13}{4} when ± is plus. Add -5 to 13.

$$x=\frac{8}{4}$$

Divide 8 by 4.

$$x=2$$

Now solve the equation x=\frac{-5±13}{4} when ± is minus. Subtract 13 from -5.

$$x=-\frac{18}{4}$$

Reduce the fraction \frac{-18}{4} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

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

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, the least common multiple of x,2.

$$2xx+2x\times 3=2\times 9+2x\times \frac{1}{2}$$

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

$$2x^{2}+2x\times 3=2\times 9+2x\times \frac{1}{2}$$

Multiply 2 and 3 to get 6.

$$2x^{2}+6x=2\times 9+2x\times \frac{1}{2}$$

Multiply 2 and 9 to get 18.

$$2x^{2}+6x=18+2x\times \frac{1}{2}$$

Cancel out 2 and 2.

$$2x^{2}+6x=18+x$$

Subtract x from both sides.

$$2x^{2}+6x-x=18$$

Combine 6x and -x to get 5x.

$$2x^{2}+5x=18$$

Divide both sides by 2.

$$\frac{2x^{2}+5x}{2}=\frac{18}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{5}{2}x=\frac{18}{2}$$

Divide 18 by 2.

$$x^{2}+\frac{5}{2}x=9$$

Divide \frac{5}{2}, the coefficient of the x term, by 2 to get \frac{5}{4}. Then add the square of \frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=9+\left(\frac{5}{4}\right)^{2}$$

Square \frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=9+\frac{25}{16}$$

Add 9 to \frac{25}{16}.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{169}{16}$$

Factor x^{2}+\frac{5}{2}x+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{169}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{169}{16}}$$

Simplify.

$$x+\frac{5}{4}=\frac{13}{4} x+\frac{5}{4}=-\frac{13}{4}$$

Subtract \frac{5}{4} from both sides of the equation.

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