Solve 150x-x^2=(1-0.8832)100*50

$$150x- { x }^{ 2 } =(1-0.8832)100 \times 50$$

Solve for x

$$x=4$$

$$x=146$$

Steps Using Factoring By Grouping

Subtract 0.8832 from 1 to get 0.1168.

$$150x-x^{2}=0.1168\times 100\times 50$$

Multiply 0.1168 and 100 to get 11.68.

$$150x-x^{2}=11.68\times 50$$

Multiply 11.68 and 50 to get 584.

$$150x-x^{2}=584$$

Subtract 584 from both sides.

$$150x-x^{2}-584=0$$

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

$$-x^{2}+150x-584=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-584. To find a and b, set up a system to be solved.

$$a+b=150 ab=-\left(-584\right)=584$$

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 584.

$$1,584 2,292 4,146 8,73$$

Calculate the sum for each pair.

$$1+584=585 2+292=294 4+146=150 8+73=81$$

The solution is the pair that gives sum 150.

$$a=146 b=4$$

Rewrite -x^{2}+150x-584 as \left(-x^{2}+146x\right)+\left(4x-584\right).

$$\left(-x^{2}+146x\right)+\left(4x-584\right)$$

Factor out -x in the first and 4 in the second group.

$$-x\left(x-146\right)+4\left(x-146\right)$$

Factor out common term x-146 by using distributive property.

$$\left(x-146\right)\left(-x+4\right)$$

To find equation solutions, solve x-146=0 and -x+4=0.

$$x=146 x=4$$

Steps Using the Quadratic Formula

Subtract 0.8832 from 1 to get 0.1168.

$$150x-x^{2}=0.1168\times 100\times 50$$

Multiply 0.1168 and 100 to get 11.68.

$$150x-x^{2}=11.68\times 50$$

Multiply 11.68 and 50 to get 584.

$$150x-x^{2}=584$$

Subtract 584 from both sides.

$$150x-x^{2}-584=0$$

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.

$$-x^{2}+150x-584=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 150 for b, and -584 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-150±\sqrt{150^{2}-4\left(-1\right)\left(-584\right)}}{2\left(-1\right)}$$

Square 150.

$$x=\frac{-150±\sqrt{22500-4\left(-1\right)\left(-584\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-150±\sqrt{22500+4\left(-584\right)}}{2\left(-1\right)}$$

Multiply 4 times -584.

$$x=\frac{-150±\sqrt{22500-2336}}{2\left(-1\right)}$$

Add 22500 to -2336.

$$x=\frac{-150±\sqrt{20164}}{2\left(-1\right)}$$

Take the square root of 20164.

$$x=\frac{-150±142}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-150±142}{-2}$$

Now solve the equation x=\frac{-150±142}{-2} when ± is plus. Add -150 to 142.

$$x=-\frac{8}{-2}$$

Divide -8 by -2.

$$x=4$$

Now solve the equation x=\frac{-150±142}{-2} when ± is minus. Subtract 142 from -150.

$$x=-\frac{292}{-2}$$

Divide -292 by -2.

$$x=146$$

The equation is now solved.

$$x=4 x=146$$

Steps for Completing the Square

Subtract 0.8832 from 1 to get 0.1168.

$$150x-x^{2}=0.1168\times 100\times 50$$

Multiply 0.1168 and 100 to get 11.68.

$$150x-x^{2}=11.68\times 50$$

Multiply 11.68 and 50 to get 584.

$$150x-x^{2}=584$$

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.

$$-x^{2}+150x=584$$

Divide both sides by -1.

$$\frac{-x^{2}+150x}{-1}=\frac{584}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{150}{-1}x=\frac{584}{-1}$$

Divide 150 by -1.

$$x^{2}-150x=\frac{584}{-1}$$

Divide 584 by -1.

$$x^{2}-150x=-584$$

Divide -150, the coefficient of the x term, by 2 to get -75. Then add the square of -75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-150x+\left(-75\right)^{2}=-584+\left(-75\right)^{2}$$

Square -75.

$$x^{2}-150x+5625=-584+5625$$

Add -584 to 5625.

$$x^{2}-150x+5625=5041$$

Factor x^{2}-150x+5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-75\right)^{2}=5041$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-75\right)^{2}}=\sqrt{5041}$$

Simplify.

$$x-75=71 x-75=-71$$

Add 75 to both sides of the equation.

$$x=146 x=4$$