Solve x-5/4=6/x

$$\frac { x - 5 } { 4 } = \frac { 6 } { x }$$

Solve for x

$$x=-3$$

$$x=8$$

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 4x, the least common multiple of 4,x.

$$x\left(x-5\right)=4\times 6$$

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

$$x^{2}-5x=4\times 6$$

Multiply 4 and 6 to get 24.

$$x^{2}-5x=24$$

Subtract 24 from both sides.

$$x^{2}-5x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-24\right)}}{2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$x=\frac{-\left(-5\right)±\sqrt{25+96}}{2}$$

Add 25 to 96.

$$x=\frac{-\left(-5\right)±\sqrt{121}}{2}$$

Take the square root of 121.

$$x=\frac{-\left(-5\right)±11}{2}$$

The opposite of -5 is 5.

$$x=\frac{5±11}{2}$$

Now solve the equation x=\frac{5±11}{2} when ± is plus. Add 5 to 11.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{5±11}{2} when ± is minus. Subtract 11 from 5.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

The equation is now solved.

$$x=8 x=-3$$

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 4x, the least common multiple of 4,x.

$$x\left(x-5\right)=4\times 6$$

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

$$x^{2}-5x=4\times 6$$

Multiply 4 and 6 to get 24.

$$x^{2}-5x=24$$

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.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=24+\left(-\frac{5}{2}\right)^{2}$$

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

$$x^{2}-5x+\frac{25}{4}=24+\frac{25}{4}$$

Add 24 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{121}{4}$$

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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{11}{2} x-\frac{5}{2}=-\frac{11}{2}$$

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

$$x=8 x=-3$$