Solve 100+x^2-20x=81x

$$100 + x ^ { 2 } - 20 x = 81 x$$

Solve for x

$$x=1$$

$$x=100$$

Steps Using Factoring

Subtract 81x from both sides.

$$100+x^{2}-20x-81x=0$$

Combine -20x and -81x to get -101x.

$$100+x^{2}-101x=0$$

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

$$x^{2}-101x+100=0$$

To solve the equation, factor x^{2}-101x+100 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-101 ab=100$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 100.

$$-1,-100 -2,-50 -4,-25 -5,-20 -10,-10$$

Calculate the sum for each pair.

$$-1-100=-101 -2-50=-52 -4-25=-29 -5-20=-25 -10-10=-20$$

The solution is the pair that gives sum -101.

$$a=-100 b=-1$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-100\right)\left(x-1\right)$$

To find equation solutions, solve x-100=0 and x-1=0.

$$x=100 x=1$$

Steps Using Factoring By Grouping

Subtract 81x from both sides.

$$100+x^{2}-20x-81x=0$$

Combine -20x and -81x to get -101x.

$$100+x^{2}-101x=0$$

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

$$x^{2}-101x+100=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+100. To find a and b, set up a system to be solved.

$$a+b=-101 ab=1\times 100=100$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 100.

$$-1,-100 -2,-50 -4,-25 -5,-20 -10,-10$$

Calculate the sum for each pair.

$$-1-100=-101 -2-50=-52 -4-25=-29 -5-20=-25 -10-10=-20$$

The solution is the pair that gives sum -101.

$$a=-100 b=-1$$

Rewrite x^{2}-101x+100 as \left(x^{2}-100x\right)+\left(-x+100\right).

$$\left(x^{2}-100x\right)+\left(-x+100\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(x-100\right)-\left(x-100\right)$$

Factor out common term x-100 by using distributive property.

$$\left(x-100\right)\left(x-1\right)$$

To find equation solutions, solve x-100=0 and x-1=0.

$$x=100 x=1$$

Steps Using the Quadratic Formula

Subtract 81x from both sides.

$$100+x^{2}-20x-81x=0$$

Combine -20x and -81x to get -101x.

$$100+x^{2}-101x=0$$

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.

$$x^{2}-101x+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -101 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-101\right)±\sqrt{\left(-101\right)^{2}-4\times 100}}{2}$$

Square -101.

$$x=\frac{-\left(-101\right)±\sqrt{10201-4\times 100}}{2}$$

Multiply -4 times 100.

$$x=\frac{-\left(-101\right)±\sqrt{10201-400}}{2}$$

Add 10201 to -400.

$$x=\frac{-\left(-101\right)±\sqrt{9801}}{2}$$

Take the square root of 9801.

$$x=\frac{-\left(-101\right)±99}{2}$$

The opposite of -101 is 101.

$$x=\frac{101±99}{2}$$

Now solve the equation x=\frac{101±99}{2} when ± is plus. Add 101 to 99.

$$x=\frac{200}{2}$$

Divide 200 by 2.

$$x=100$$

Now solve the equation x=\frac{101±99}{2} when ± is minus. Subtract 99 from 101.

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

Divide 2 by 2.

$$x=1$$

The equation is now solved.

$$x=100 x=1$$

Steps for Completing the Square

Subtract 81x from both sides.

$$100+x^{2}-20x-81x=0$$

Combine -20x and -81x to get -101x.

$$100+x^{2}-101x=0$$

Subtract 100 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-101x=-100$$

Divide -101, the coefficient of the x term, by 2 to get -\frac{101}{2}. Then add the square of -\frac{101}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-101x+\left(-\frac{101}{2}\right)^{2}=-100+\left(-\frac{101}{2}\right)^{2}$$

Square -\frac{101}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-101x+\frac{10201}{4}=-100+\frac{10201}{4}$$

Add -100 to \frac{10201}{4}.

$$x^{2}-101x+\frac{10201}{4}=\frac{9801}{4}$$

Factor x^{2}-101x+\frac{10201}{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}.

$$\left(x-\frac{101}{2}\right)^{2}=\frac{9801}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{101}{2}\right)^{2}}=\sqrt{\frac{9801}{4}}$$

Simplify.

$$x-\frac{101}{2}=\frac{99}{2} x-\frac{101}{2}=-\frac{99}{2}$$

Add \frac{101}{2} to both sides of the equation.

$$x=100 x=1$$