Solve 4x^2/(5-x)(4-x)=322

$$\frac { 4 x ^ { 2 } } { ( 5 - x ) ( 4 - x ) } = 322$$

Solve for x

$$x=\frac{\sqrt{51681}}{318}+\frac{483}{106}\approx 5.271492312$$

$$x=-\frac{\sqrt{51681}}{318}+\frac{483}{106}\approx 3.841715235$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 4,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x-4\right).

$$4x^{2}=322\left(x-5\right)\left(x-4\right)$$

Use the distributive property to multiply 322 by x-5.

$$4x^{2}=\left(322x-1610\right)\left(x-4\right)$$

Use the distributive property to multiply 322x-1610 by x-4 and combine like terms.

$$4x^{2}=322x^{2}-2898x+6440$$

Subtract 322x^{2} from both sides.

$$4x^{2}-322x^{2}=-2898x+6440$$

Combine 4x^{2} and -322x^{2} to get -318x^{2}.

$$-318x^{2}=-2898x+6440$$

Add 2898x to both sides.

$$-318x^{2}+2898x=6440$$

Subtract 6440 from both sides.

$$-318x^{2}+2898x-6440=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -318 for a, 2898 for b, and -6440 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2898±\sqrt{2898^{2}-4\left(-318\right)\left(-6440\right)}}{2\left(-318\right)}$$

Square 2898.

$$x=\frac{-2898±\sqrt{8398404-4\left(-318\right)\left(-6440\right)}}{2\left(-318\right)}$$

Multiply -4 times -318.

$$x=\frac{-2898±\sqrt{8398404+1272\left(-6440\right)}}{2\left(-318\right)}$$

Multiply 1272 times -6440.

$$x=\frac{-2898±\sqrt{8398404-8191680}}{2\left(-318\right)}$$

Add 8398404 to -8191680.

$$x=\frac{-2898±\sqrt{206724}}{2\left(-318\right)}$$

Take the square root of 206724.

$$x=\frac{-2898±2\sqrt{51681}}{2\left(-318\right)}$$

Multiply 2 times -318.

$$x=\frac{-2898±2\sqrt{51681}}{-636}$$

Now solve the equation x=\frac{-2898±2\sqrt{51681}}{-636} when ± is plus. Add -2898 to 2\sqrt{51681}.

$$x=\frac{2\sqrt{51681}-2898}{-636}$$

Divide -2898+2\sqrt{51681} by -636.

$$x=-\frac{\sqrt{51681}}{318}+\frac{483}{106}$$

Now solve the equation x=\frac{-2898±2\sqrt{51681}}{-636} when ± is minus. Subtract 2\sqrt{51681} from -2898.

$$x=\frac{-2\sqrt{51681}-2898}{-636}$$

Divide -2898-2\sqrt{51681} by -636.

$$x=\frac{\sqrt{51681}}{318}+\frac{483}{106}$$

The equation is now solved.

$$x=-\frac{\sqrt{51681}}{318}+\frac{483}{106} x=\frac{\sqrt{51681}}{318}+\frac{483}{106}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 4,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x-4\right).

$$4x^{2}=322\left(x-5\right)\left(x-4\right)$$

Use the distributive property to multiply 322 by x-5.

$$4x^{2}=\left(322x-1610\right)\left(x-4\right)$$

Use the distributive property to multiply 322x-1610 by x-4 and combine like terms.

$$4x^{2}=322x^{2}-2898x+6440$$

Subtract 322x^{2} from both sides.

$$4x^{2}-322x^{2}=-2898x+6440$$

Combine 4x^{2} and -322x^{2} to get -318x^{2}.

$$-318x^{2}=-2898x+6440$$

Add 2898x to both sides.

$$-318x^{2}+2898x=6440$$

Divide both sides by -318.

$$\frac{-318x^{2}+2898x}{-318}=\frac{6440}{-318}$$

Dividing by -318 undoes the multiplication by -318.

$$x^{2}+\frac{2898}{-318}x=\frac{6440}{-318}$$

Reduce the fraction \frac{2898}{-318} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{483}{53}x=\frac{6440}{-318}$$

Reduce the fraction \frac{6440}{-318} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{483}{53}x=-\frac{3220}{159}$$

Divide -\frac{483}{53}, the coefficient of the x term, by 2 to get -\frac{483}{106}. Then add the square of -\frac{483}{106} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{483}{53}x+\left(-\frac{483}{106}\right)^{2}=-\frac{3220}{159}+\left(-\frac{483}{106}\right)^{2}$$

Square -\frac{483}{106} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{483}{53}x+\frac{233289}{11236}=-\frac{3220}{159}+\frac{233289}{11236}$$

Add -\frac{3220}{159} to \frac{233289}{11236} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{483}{53}x+\frac{233289}{11236}=\frac{17227}{33708}$$

Factor x^{2}-\frac{483}{53}x+\frac{233289}{11236}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{483}{106}\right)^{2}=\frac{17227}{33708}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{483}{106}\right)^{2}}=\sqrt{\frac{17227}{33708}}$$

Simplify.

$$x-\frac{483}{106}=\frac{\sqrt{51681}}{318} x-\frac{483}{106}=-\frac{\sqrt{51681}}{318}$$

Add \frac{483}{106} to both sides of the equation.

$$x=\frac{\sqrt{51681}}{318}+\frac{483}{106} x=-\frac{\sqrt{51681}}{318}+\frac{483}{106}$$