Solve 2x-3/5-xx=4

$$\frac{ 2x-3 }{ 5-x } x=4$$

Solve for x

$$x = \frac{\sqrt{161} - 1}{4} \approx 2.922144385$$

$$x=\frac{-\sqrt{161}-1}{4}\approx -3.422144385$$

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 -x+5.

$$\left(2x-3\right)x=4\left(-x+5\right)$$

Use the distributive property to multiply 2x-3 by x.

$$2x^{2}-3x=4\left(-x+5\right)$$

Use the distributive property to multiply 4 by -x+5.

$$2x^{2}-3x=-4x+20$$

Add 4x to both sides.

$$2x^{2}-3x+4x=20$$

Combine -3x and 4x to get x.

$$2x^{2}+x=20$$

Subtract 20 from both sides.

$$2x^{2}+x-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 1 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 2\left(-20\right)}}{2\times 2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 2\left(-20\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-1±\sqrt{1-8\left(-20\right)}}{2\times 2}$$

Multiply -8 times -20.

$$x=\frac{-1±\sqrt{1+160}}{2\times 2}$$

Add 1 to 160.

$$x=\frac{-1±\sqrt{161}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-1±\sqrt{161}}{4}$$

Now solve the equation x=\frac{-1±\sqrt{161}}{4} when ± is plus. Add -1 to \sqrt{161}.

$$x=\frac{\sqrt{161}-1}{4}$$

Now solve the equation x=\frac{-1±\sqrt{161}}{4} when ± is minus. Subtract \sqrt{161} from -1.

$$x=\frac{-\sqrt{161}-1}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{161}-1}{4} x=\frac{-\sqrt{161}-1}{4}$$

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 -x+5.

$$\left(2x-3\right)x=4\left(-x+5\right)$$

Use the distributive property to multiply 2x-3 by x.

$$2x^{2}-3x=4\left(-x+5\right)$$

Use the distributive property to multiply 4 by -x+5.

$$2x^{2}-3x=-4x+20$$

Add 4x to both sides.

$$2x^{2}-3x+4x=20$$

Combine -3x and 4x to get x.

$$2x^{2}+x=20$$

Divide both sides by 2.

$$\frac{2x^{2}+x}{2}=\frac{20}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{1}{2}x=\frac{20}{2}$$

Divide 20 by 2.

$$x^{2}+\frac{1}{2}x=10$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=10+\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=10+\frac{1}{16}$$

Add 10 to \frac{1}{16}.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{161}{16}$$

Factor x^{2}+\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{161}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{161}{16}}$$

Simplify.

$$x+\frac{1}{4}=\frac{\sqrt{161}}{4} x+\frac{1}{4}=-\frac{\sqrt{161}}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

$$x=\frac{\sqrt{161}-1}{4} x=\frac{-\sqrt{161}-1}{4}$$