Solve x+15/4-21/x+2=2

$$\frac { x + 15 } { 4 } - \frac { 21 } { x + 2 } = 2$$

Solve for x

$$x=-14$$

$$x=5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by 4\left(x+2\right), the least common multiple of 4,x+2.

$$\left(x+2\right)\left(x+15\right)-4\times 21=8\left(x+2\right)$$

Use the distributive property to multiply x+2 by x+15 and combine like terms.

$$x^{2}+17x+30-4\times 21=8\left(x+2\right)$$

Multiply -4 and 21 to get -84.

$$x^{2}+17x+30-84=8\left(x+2\right)$$

Subtract 84 from 30 to get -54.

$$x^{2}+17x-54=8\left(x+2\right)$$

Use the distributive property to multiply 8 by x+2.

$$x^{2}+17x-54=8x+16$$

Subtract 8x from both sides.

$$x^{2}+17x-54-8x=16$$

Combine 17x and -8x to get 9x.

$$x^{2}+9x-54=16$$

Subtract 16 from both sides.

$$x^{2}+9x-54-16=0$$

Subtract 16 from -54 to get -70.

$$x^{2}+9x-70=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 9 for b, and -70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-70\right)}}{2}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-70\right)}}{2}$$

Multiply -4 times -70.

$$x=\frac{-9±\sqrt{81+280}}{2}$$

Add 81 to 280.

$$x=\frac{-9±\sqrt{361}}{2}$$

Take the square root of 361.

$$x=\frac{-9±19}{2}$$

Now solve the equation x=\frac{-9±19}{2} when ± is plus. Add -9 to 19.

$$x=\frac{10}{2}$$

Divide 10 by 2.

$$x=5$$

Now solve the equation x=\frac{-9±19}{2} when ± is minus. Subtract 19 from -9.

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

Divide -28 by 2.

$$x=-14$$

The equation is now solved.

$$x=5 x=-14$$

Steps for Completing the Square

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by 4\left(x+2\right), the least common multiple of 4,x+2.

$$\left(x+2\right)\left(x+15\right)-4\times 21=8\left(x+2\right)$$

Use the distributive property to multiply x+2 by x+15 and combine like terms.

$$x^{2}+17x+30-4\times 21=8\left(x+2\right)$$

Multiply -4 and 21 to get -84.

$$x^{2}+17x+30-84=8\left(x+2\right)$$

Subtract 84 from 30 to get -54.

$$x^{2}+17x-54=8\left(x+2\right)$$

Use the distributive property to multiply 8 by x+2.

$$x^{2}+17x-54=8x+16$$

Subtract 8x from both sides.

$$x^{2}+17x-54-8x=16$$

Combine 17x and -8x to get 9x.

$$x^{2}+9x-54=16$$

Add 54 to both sides.

$$x^{2}+9x=16+54$$

Add 16 and 54 to get 70.

$$x^{2}+9x=70$$

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.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=70+\left(\frac{9}{2}\right)^{2}$$

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

$$x^{2}+9x+\frac{81}{4}=70+\frac{81}{4}$$

Add 70 to \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{361}{4}$$

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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{361}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{19}{2} x+\frac{9}{2}=-\frac{19}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=5 x=-14$$