Solve x-9/3=56/x+4

$$\frac { x - 9 } { 3 } = \frac { 56 } { x + 4 }$$

Solve for x

$$x=-12$$

$$x=17$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by 3\left(x+4\right), the least common multiple of 3,x+4.

$$\left(x+4\right)\left(x-9\right)=3\times 56$$

Use the distributive property to multiply x+4 by x-9 and combine like terms.

$$x^{2}-5x-36=3\times 56$$

Multiply 3 and 56 to get 168.

$$x^{2}-5x-36=168$$

Subtract 168 from both sides.

$$x^{2}-5x-36-168=0$$

Subtract 168 from -36 to get -204.

$$x^{2}-5x-204=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -204 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-204\right)}}{2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-204\right)}}{2}$$

Multiply -4 times -204.

$$x=\frac{-\left(-5\right)±\sqrt{25+816}}{2}$$

Add 25 to 816.

$$x=\frac{-\left(-5\right)±\sqrt{841}}{2}$$

Take the square root of 841.

$$x=\frac{-\left(-5\right)±29}{2}$$

The opposite of -5 is 5.

$$x=\frac{5±29}{2}$$

Now solve the equation x=\frac{5±29}{2} when ± is plus. Add 5 to 29.

$$x=\frac{34}{2}$$

Divide 34 by 2.

$$x=17$$

Now solve the equation x=\frac{5±29}{2} when ± is minus. Subtract 29 from 5.

$$x=-\frac{24}{2}$$

Divide -24 by 2.

$$x=-12$$

The equation is now solved.

$$x=17 x=-12$$

Steps for Completing the Square

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by 3\left(x+4\right), the least common multiple of 3,x+4.

$$\left(x+4\right)\left(x-9\right)=3\times 56$$

Use the distributive property to multiply x+4 by x-9 and combine like terms.

$$x^{2}-5x-36=3\times 56$$

Multiply 3 and 56 to get 168.

$$x^{2}-5x-36=168$$

Add 36 to both sides.

$$x^{2}-5x=168+36$$

Add 168 and 36 to get 204.

$$x^{2}-5x=204$$

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

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=204+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=204+\frac{25}{4}$$

Add 204 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{841}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{841}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{841}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{29}{2} x-\frac{5}{2}=-\frac{29}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=17 x=-12$$