Solve 4/x+5=x/9

$$\frac { 4 } { x + 5 } = \frac { x } { 9 }$$

Solve for x

$$x=-9$$

$$x=4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by 9\left(x+5\right), the least common multiple of x+5,9.

$$9\times 4=\left(x+5\right)x$$

Multiply 9 and 4 to get 36.

$$36=\left(x+5\right)x$$

Use the distributive property to multiply x+5 by x.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 36 from both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-36\right)}}{2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-36\right)}}{2}$$

Multiply -4 times -36.

$$x=\frac{-5±\sqrt{25+144}}{2}$$

Add 25 to 144.

$$x=\frac{-5±\sqrt{169}}{2}$$

Take the square root of 169.

$$x=\frac{-5±13}{2}$$

Now solve the equation x=\frac{-5±13}{2} when ± is plus. Add -5 to 13.

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

Now solve the equation x=\frac{-5±13}{2} when ± is minus. Subtract 13 from -5.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=4 x=-9$$

Steps for Completing the Square

Variable x cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by 9\left(x+5\right), the least common multiple of x+5,9.

$$9\times 4=\left(x+5\right)x$$

Multiply 9 and 4 to get 36.

$$36=\left(x+5\right)x$$

Use the distributive property to multiply x+5 by x.

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

Swap sides so that all variable terms are on the left hand side.

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

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.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=36+\left(\frac{5}{2}\right)^{2}$$

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

$$x^{2}+5x+\frac{25}{4}=36+\frac{25}{4}$$

Add 36 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{169}{4}$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+\frac{5}{2}=\frac{13}{2} x+\frac{5}{2}=-\frac{13}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=4 x=-9$$