Solve 4-2x/4-x=x-5/x+4

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

Solve for x

$$x = \frac{\sqrt{313} - 13}{2} \approx 2.345903006$$

$$x=\frac{-\sqrt{313}-13}{2}\approx -15.345903006$$

Steps Using the Quadratic Formula

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

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

To find the opposite of 4+x, find the opposite of each term.

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

Use the distributive property to multiply -4-x by 4-2x and combine like terms.

$$-16+4x+2x^{2}=\left(x-4\right)\left(x-5\right)$$

Use the distributive property to multiply x-4 by x-5 and combine like terms.

$$-16+4x+2x^{2}=x^{2}-9x+20$$

Subtract x^{2} from both sides.

$$-16+4x+2x^{2}-x^{2}=-9x+20$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$-16+4x+x^{2}=-9x+20$$

Add 9x to both sides.

$$-16+4x+x^{2}+9x=20$$

Combine 4x and 9x to get 13x.

$$-16+13x+x^{2}=20$$

Subtract 20 from both sides.

$$-16+13x+x^{2}-20=0$$

Subtract 20 from -16 to get -36.

$$-36+13x+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+13x-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 13 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-36\right)}}{2}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-36\right)}}{2}$$

Multiply -4 times -36.

$$x=\frac{-13±\sqrt{169+144}}{2}$$

Add 169 to 144.

$$x=\frac{-13±\sqrt{313}}{2}$$

Now solve the equation x=\frac{-13±\sqrt{313}}{2} when ± is plus. Add -13 to \sqrt{313}.

$$x=\frac{\sqrt{313}-13}{2}$$

Now solve the equation x=\frac{-13±\sqrt{313}}{2} when ± is minus. Subtract \sqrt{313} from -13.

$$x=\frac{-\sqrt{313}-13}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{313}-13}{2} x=\frac{-\sqrt{313}-13}{2}$$

Steps for Completing the Square

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

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

To find the opposite of 4+x, find the opposite of each term.

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

Use the distributive property to multiply -4-x by 4-2x and combine like terms.

$$-16+4x+2x^{2}=\left(x-4\right)\left(x-5\right)$$

Use the distributive property to multiply x-4 by x-5 and combine like terms.

$$-16+4x+2x^{2}=x^{2}-9x+20$$

Subtract x^{2} from both sides.

$$-16+4x+2x^{2}-x^{2}=-9x+20$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$-16+4x+x^{2}=-9x+20$$

Add 9x to both sides.

$$-16+4x+x^{2}+9x=20$$

Combine 4x and 9x to get 13x.

$$-16+13x+x^{2}=20$$

Add 16 to both sides.

$$13x+x^{2}=20+16$$

Add 20 and 16 to get 36.

$$13x+x^{2}=36$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+13x=36$$

Divide 13, the coefficient of the x term, by 2 to get \frac{13}{2}. Then add the square of \frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+13x+\left(\frac{13}{2}\right)^{2}=36+\left(\frac{13}{2}\right)^{2}$$

Square \frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+13x+\frac{169}{4}=36+\frac{169}{4}$$

Add 36 to \frac{169}{4}.

$$x^{2}+13x+\frac{169}{4}=\frac{313}{4}$$

Factor x^{2}+13x+\frac{169}{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}.

$$\left(x+\frac{13}{2}\right)^{2}=\frac{313}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{2}\right)^{2}}=\sqrt{\frac{313}{4}}$$

Simplify.

$$x+\frac{13}{2}=\frac{\sqrt{313}}{2} x+\frac{13}{2}=-\frac{\sqrt{313}}{2}$$

Subtract \frac{13}{2} from both sides of the equation.

$$x=\frac{\sqrt{313}-13}{2} x=\frac{-\sqrt{313}-13}{2}$$