Solve 4x^2/x^2+25-10x+2x-24x-120/x-5=0

$$\frac { 4 x ^ { 2 } } { x ^ { 2 } + 25 - 10 x } + \frac { 2 x - 24 x - 120 } { x - 5 } = 0$$

Solve for x

$$x = \frac{5 \sqrt{433} - 5}{18} \approx 5.502403346$$

$$x=\frac{-5\sqrt{433}-5}{18}\approx -6.057958902$$

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 \left(x-5\right)^{2}, the least common multiple of x^{2}+25-10x,x-5.

$$4x^{2}+\left(x-5\right)\left(2x-24x-120\right)=0$$

Combine 2x and -24x to get -22x.

$$4x^{2}+\left(x-5\right)\left(-22x-120\right)=0$$

Use the distributive property to multiply x-5 by -22x-120 and combine like terms.

$$4x^{2}-22x^{2}-10x+600=0$$

Combine 4x^{2} and -22x^{2} to get -18x^{2}.

$$-18x^{2}-10x+600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -18 for a, -10 for b, and 600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-18\right)\times 600}}{2\left(-18\right)}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-18\right)\times 600}}{2\left(-18\right)}$$

Multiply -4 times -18.

$$x=\frac{-\left(-10\right)±\sqrt{100+72\times 600}}{2\left(-18\right)}$$

Multiply 72 times 600.

$$x=\frac{-\left(-10\right)±\sqrt{100+43200}}{2\left(-18\right)}$$

Add 100 to 43200.

$$x=\frac{-\left(-10\right)±\sqrt{43300}}{2\left(-18\right)}$$

Take the square root of 43300.

$$x=\frac{-\left(-10\right)±10\sqrt{433}}{2\left(-18\right)}$$

The opposite of -10 is 10.

$$x=\frac{10±10\sqrt{433}}{2\left(-18\right)}$$

Multiply 2 times -18.

$$x=\frac{10±10\sqrt{433}}{-36}$$

Now solve the equation x=\frac{10±10\sqrt{433}}{-36} when ± is plus. Add 10 to 10\sqrt{433}.

$$x=\frac{10\sqrt{433}+10}{-36}$$

Divide 10+10\sqrt{433} by -36.

$$x=\frac{-5\sqrt{433}-5}{18}$$

Now solve the equation x=\frac{10±10\sqrt{433}}{-36} when ± is minus. Subtract 10\sqrt{433} from 10.

$$x=\frac{10-10\sqrt{433}}{-36}$$

Divide 10-10\sqrt{433} by -36.

$$x=\frac{5\sqrt{433}-5}{18}$$

The equation is now solved.

$$x=\frac{-5\sqrt{433}-5}{18} x=\frac{5\sqrt{433}-5}{18}$$

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 \left(x-5\right)^{2}, the least common multiple of x^{2}+25-10x,x-5.

$$4x^{2}+\left(x-5\right)\left(2x-24x-120\right)=0$$

Combine 2x and -24x to get -22x.

$$4x^{2}+\left(x-5\right)\left(-22x-120\right)=0$$

Use the distributive property to multiply x-5 by -22x-120 and combine like terms.

$$4x^{2}-22x^{2}-10x+600=0$$

Combine 4x^{2} and -22x^{2} to get -18x^{2}.

$$-18x^{2}-10x+600=0$$

Subtract 600 from both sides. Anything subtracted from zero gives its negation.

$$-18x^{2}-10x=-600$$

Divide both sides by -18.

$$\frac{-18x^{2}-10x}{-18}=-\frac{600}{-18}$$

Dividing by -18 undoes the multiplication by -18.

$$x^{2}+\left(-\frac{10}{-18}\right)x=-\frac{600}{-18}$$

Reduce the fraction \frac{-10}{-18} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{5}{9}x=-\frac{600}{-18}$$

Reduce the fraction \frac{-600}{-18} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{5}{9}x=\frac{100}{3}$$

Divide \frac{5}{9}, the coefficient of the x term, by 2 to get \frac{5}{18}. Then add the square of \frac{5}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{9}x+\left(\frac{5}{18}\right)^{2}=\frac{100}{3}+\left(\frac{5}{18}\right)^{2}$$

Square \frac{5}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{9}x+\frac{25}{324}=\frac{100}{3}+\frac{25}{324}$$

Add \frac{100}{3} to \frac{25}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{9}x+\frac{25}{324}=\frac{10825}{324}$$

Factor x^{2}+\frac{5}{9}x+\frac{25}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{18}\right)^{2}=\frac{10825}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{18}\right)^{2}}=\sqrt{\frac{10825}{324}}$$

Simplify.

$$x+\frac{5}{18}=\frac{5\sqrt{433}}{18} x+\frac{5}{18}=-\frac{5\sqrt{433}}{18}$$

Subtract \frac{5}{18} from both sides of the equation.

$$x=\frac{5\sqrt{433}-5}{18} x=\frac{-5\sqrt{433}-5}{18}$$