Solve 5x-16/9+3-x/2=1/x+12

$$\frac{ 5x-16 }{ 9 } + \frac{ 3-x }{ 2 } = \frac{ 1 }{ x+12 }$$

Solve for x

$$x=-13$$

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -12 since division by zero is not defined. Multiply both sides of the equation by 18\left(x+12\right), the least common multiple of 9,2,x+12.

$$\left(2x+24\right)\left(5x-16\right)+\left(9x+108\right)\left(3-x\right)=18$$

Use the distributive property to multiply 2x+24 by 5x-16 and combine like terms.

$$10x^{2}+88x-384+\left(9x+108\right)\left(3-x\right)=18$$

Use the distributive property to multiply 9x+108 by 3-x and combine like terms.

$$10x^{2}+88x-384-81x-9x^{2}+324=18$$

Combine 88x and -81x to get 7x.

$$10x^{2}+7x-384-9x^{2}+324=18$$

Combine 10x^{2} and -9x^{2} to get x^{2}.

$$x^{2}+7x-384+324=18$$

Add -384 and 324 to get -60.

$$x^{2}+7x-60=18$$

Subtract 18 from both sides.

$$x^{2}+7x-60-18=0$$

Subtract 18 from -60 to get -78.

$$x^{2}+7x-78=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 7 for b, and -78 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-78\right)}}{2}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-78\right)}}{2}$$

Multiply -4 times -78.

$$x=\frac{-7±\sqrt{49+312}}{2}$$

Add 49 to 312.

$$x=\frac{-7±\sqrt{361}}{2}$$

Take the square root of 361.

$$x=\frac{-7±19}{2}$$

Now solve the equation x=\frac{-7±19}{2} when ± is plus. Add -7 to 19.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Now solve the equation x=\frac{-7±19}{2} when ± is minus. Subtract 19 from -7.

$$x=-\frac{26}{2}$$

Divide -26 by 2.

$$x=-13$$

The equation is now solved.

$$x=6 x=-13$$

Steps for Completing the Square

Variable x cannot be equal to -12 since division by zero is not defined. Multiply both sides of the equation by 18\left(x+12\right), the least common multiple of 9,2,x+12.

$$\left(2x+24\right)\left(5x-16\right)+\left(9x+108\right)\left(3-x\right)=18$$

Use the distributive property to multiply 2x+24 by 5x-16 and combine like terms.

$$10x^{2}+88x-384+\left(9x+108\right)\left(3-x\right)=18$$

Use the distributive property to multiply 9x+108 by 3-x and combine like terms.

$$10x^{2}+88x-384-81x-9x^{2}+324=18$$

Combine 88x and -81x to get 7x.

$$10x^{2}+7x-384-9x^{2}+324=18$$

Combine 10x^{2} and -9x^{2} to get x^{2}.

$$x^{2}+7x-384+324=18$$

Add -384 and 324 to get -60.

$$x^{2}+7x-60=18$$

Add 60 to both sides.

$$x^{2}+7x=18+60$$

Add 18 and 60 to get 78.

$$x^{2}+7x=78$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=78+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+7x+\frac{49}{4}=78+\frac{49}{4}$$

Add 78 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{361}{4}$$

Factor x^{2}+7x+\frac{49}{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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{361}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{19}{2} x+\frac{7}{2}=-\frac{19}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=6 x=-13$$