Solve (5x-2)(4-x)/x+10=0

$$\frac { ( 5 x - 2 ) ( 4 - x ) } { x + 10 } = 0$$

Solve for x

$$x=4$$

$$x=\frac{2}{5}=0.4$$

Steps Using Factoring By Grouping

Variable x cannot be equal to -10 since division by zero is not defined. Multiply both sides of the equation by x+10.

$$\left(5x-2\right)\left(4-x\right)=0$$

Use the distributive property to multiply 5x-2 by 4-x and combine like terms.

$$22x-5x^{2}-8=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-5x^{2}+22x-8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -5x^{2}+ax+bx-8. To find a and b, set up a system to be solved.

$$a+b=22 ab=-5\left(-8\right)=40$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 40.

$$1,40 2,20 4,10 5,8$$

Calculate the sum for each pair.

$$1+40=41 2+20=22 4+10=14 5+8=13$$

The solution is the pair that gives sum 22.

$$a=20 b=2$$

Rewrite -5x^{2}+22x-8 as \left(-5x^{2}+20x\right)+\left(2x-8\right).

$$\left(-5x^{2}+20x\right)+\left(2x-8\right)$$

Factor out 5x in the first and -2 in the second group.

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

Factor out common term -x+4 by using distributive property.

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

To find equation solutions, solve -x+4=0 and 5x-2=0.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to -10 since division by zero is not defined. Multiply both sides of the equation by x+10.

$$\left(5x-2\right)\left(4-x\right)=0$$

Use the distributive property to multiply 5x-2 by 4-x and combine like terms.

$$22x-5x^{2}-8=0$$

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.

$$-5x^{2}+22x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 22 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-22±\sqrt{22^{2}-4\left(-5\right)\left(-8\right)}}{2\left(-5\right)}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\left(-5\right)\left(-8\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-22±\sqrt{484+20\left(-8\right)}}{2\left(-5\right)}$$

Multiply 20 times -8.

$$x=\frac{-22±\sqrt{484-160}}{2\left(-5\right)}$$

Add 484 to -160.

$$x=\frac{-22±\sqrt{324}}{2\left(-5\right)}$$

Take the square root of 324.

$$x=\frac{-22±18}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-22±18}{-10}$$

Now solve the equation x=\frac{-22±18}{-10} when ± is plus. Add -22 to 18.

$$x=-\frac{4}{-10}$$

Reduce the fraction \frac{-4}{-10} to lowest terms by extracting and canceling out 2.

$$x=\frac{2}{5}$$

Now solve the equation x=\frac{-22±18}{-10} when ± is minus. Subtract 18 from -22.

$$x=-\frac{40}{-10}$$

Divide -40 by -10.

$$x=4$$

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to -10 since division by zero is not defined. Multiply both sides of the equation by x+10.

$$\left(5x-2\right)\left(4-x\right)=0$$

Use the distributive property to multiply 5x-2 by 4-x and combine like terms.

$$22x-5x^{2}-8=0$$

Add 8 to both sides. Anything plus zero gives itself.

$$22x-5x^{2}=8$$

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.

$$-5x^{2}+22x=8$$

Divide both sides by -5.

$$\frac{-5x^{2}+22x}{-5}=\frac{8}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{22}{-5}x=\frac{8}{-5}$$

Divide 22 by -5.

$$x^{2}-\frac{22}{5}x=\frac{8}{-5}$$

Divide 8 by -5.

$$x^{2}-\frac{22}{5}x=-\frac{8}{5}$$

Divide -\frac{22}{5}, the coefficient of the x term, by 2 to get -\frac{11}{5}. Then add the square of -\frac{11}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{22}{5}x+\left(-\frac{11}{5}\right)^{2}=-\frac{8}{5}+\left(-\frac{11}{5}\right)^{2}$$

Square -\frac{11}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{22}{5}x+\frac{121}{25}=-\frac{8}{5}+\frac{121}{25}$$

Add -\frac{8}{5} to \frac{121}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{22}{5}x+\frac{121}{25}=\frac{81}{25}$$

Factor x^{2}-\frac{22}{5}x+\frac{121}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{5}\right)^{2}=\frac{81}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{5}\right)^{2}}=\sqrt{\frac{81}{25}}$$

Simplify.

$$x-\frac{11}{5}=\frac{9}{5} x-\frac{11}{5}=-\frac{9}{5}$$

Add \frac{11}{5} to both sides of the equation.

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