Solve (4-2x)^2/x(4-x)=1.479

$$\frac { ( 4 - 2 x ) ^ { 2 } } { x ( 4 - x ) } = 1.479$$

Solve for x

$$x=\frac{2\sqrt{8103441}}{5479}+2\approx 3.039114565$$

$$x=-\frac{2\sqrt{8103441}}{5479}+2\approx 0.960885435$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,4 since division by zero is not defined. Multiply both sides of the equation by x\left(-x+4\right).

$$\left(4-2x\right)^{2}=1.479x\left(-x+4\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-2x\right)^{2}.

$$16-16x+4x^{2}=1.479x\left(-x+4\right)$$

Use the distributive property to multiply 1.479x by -x+4.

$$16-16x+4x^{2}=-1.479x^{2}+5.916x$$

Add 1.479x^{2} to both sides.

$$16-16x+4x^{2}+1.479x^{2}=5.916x$$

Combine 4x^{2} and 1.479x^{2} to get 5.479x^{2}.

$$16-16x+5.479x^{2}=5.916x$$

Subtract 5.916x from both sides.

$$16-16x+5.479x^{2}-5.916x=0$$

Combine -16x and -5.916x to get -21.916x.

$$16-21.916x+5.479x^{2}=0$$

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.

$$5.479x^{2}-21.916x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5.479 for a, -21.916 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21.916\right)±\sqrt{\left(-21.916\right)^{2}-4\times 5.479\times 16}}{2\times 5.479}$$

Square -21.916 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-21.916\right)±\sqrt{480.311056-4\times 5.479\times 16}}{2\times 5.479}$$

Multiply -4 times 5.479.

$$x=\frac{-\left(-21.916\right)±\sqrt{480.311056-21.916\times 16}}{2\times 5.479}$$

Multiply -21.916 times 16.

$$x=\frac{-\left(-21.916\right)±\sqrt{480.311056-350.656}}{2\times 5.479}$$

Add 480.311056 to -350.656 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-21.916\right)±\sqrt{129.655056}}{2\times 5.479}$$

Take the square root of 129.655056.

$$x=\frac{-\left(-21.916\right)±\frac{\sqrt{8103441}}{250}}{2\times 5.479}$$

The opposite of -21.916 is 21.916.

$$x=\frac{21.916±\frac{\sqrt{8103441}}{250}}{2\times 5.479}$$

Multiply 2 times 5.479.

$$x=\frac{21.916±\frac{\sqrt{8103441}}{250}}{10.958}$$

Now solve the equation x=\frac{21.916±\frac{\sqrt{8103441}}{250}}{10.958} when ± is plus. Add 21.916 to \frac{\sqrt{8103441}}{250}.

$$x=\frac{\sqrt{8103441}+5479}{10.958\times 250}$$

Divide \frac{5479+\sqrt{8103441}}{250} by 10.958 by multiplying \frac{5479+\sqrt{8103441}}{250} by the reciprocal of 10.958.

$$x=\frac{2\sqrt{8103441}}{5479}+2$$

Now solve the equation x=\frac{21.916±\frac{\sqrt{8103441}}{250}}{10.958} when ± is minus. Subtract \frac{\sqrt{8103441}}{250} from 21.916.

$$x=\frac{5479-\sqrt{8103441}}{10.958\times 250}$$

Divide \frac{5479-\sqrt{8103441}}{250} by 10.958 by multiplying \frac{5479-\sqrt{8103441}}{250} by the reciprocal of 10.958.

$$x=-\frac{2\sqrt{8103441}}{5479}+2$$

The equation is now solved.

$$x=\frac{2\sqrt{8103441}}{5479}+2 x=-\frac{2\sqrt{8103441}}{5479}+2$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,4 since division by zero is not defined. Multiply both sides of the equation by x\left(-x+4\right).

$$\left(4-2x\right)^{2}=1.479x\left(-x+4\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-2x\right)^{2}.

$$16-16x+4x^{2}=1.479x\left(-x+4\right)$$

Use the distributive property to multiply 1.479x by -x+4.

$$16-16x+4x^{2}=-1.479x^{2}+5.916x$$

Add 1.479x^{2} to both sides.

$$16-16x+4x^{2}+1.479x^{2}=5.916x$$

Combine 4x^{2} and 1.479x^{2} to get 5.479x^{2}.

$$16-16x+5.479x^{2}=5.916x$$

Subtract 5.916x from both sides.

$$16-16x+5.479x^{2}-5.916x=0$$

Combine -16x and -5.916x to get -21.916x.

$$16-21.916x+5.479x^{2}=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$-21.916x+5.479x^{2}=-16$$

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.

$$5.479x^{2}-21.916x=-16$$

Divide both sides of the equation by 5.479, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{5.479x^{2}-21.916x}{5.479}=-\frac{16}{5.479}$$

Dividing by 5.479 undoes the multiplication by 5.479.

$$x^{2}+\left(-\frac{21.916}{5.479}\right)x=-\frac{16}{5.479}$$

Divide -21.916 by 5.479 by multiplying -21.916 by the reciprocal of 5.479.

$$x^{2}-4x=-\frac{16}{5.479}$$

Divide -16 by 5.479 by multiplying -16 by the reciprocal of 5.479.

$$x^{2}-4x=-\frac{16000}{5479}$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-\frac{16000}{5479}+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-\frac{16000}{5479}+4$$

Add -\frac{16000}{5479} to 4.

$$x^{2}-4x+4=\frac{5916}{5479}$$

Factor x^{2}-4x+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}.

$$\left(x-2\right)^{2}=\frac{5916}{5479}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{5916}{5479}}$$

Simplify.

$$x-2=\frac{2\sqrt{8103441}}{5479} x-2=-\frac{2\sqrt{8103441}}{5479}$$

Add 2 to both sides of the equation.

$$x=\frac{2\sqrt{8103441}}{5479}+2 x=-\frac{2\sqrt{8103441}}{5479}+2$$