Solve x^2-x^-9900=0

$${ x }^{ 2 } - { x }^{ } -9900=0$$

Solve for x

$$x=-99$$

$$x=100$$

Steps Using Factoring

Calculate x to the power of 1 and get x.

$$x^{2}-x-9900=0$$

To solve the equation, factor x^{2}-x-9900 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.

$$a+b=-1 ab=-9900$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -9900.

$$1,-9900 2,-4950 3,-3300 4,-2475 5,-1980 6,-1650 9,-1100 10,-990 11,-900 12,-825 15,-660 18,-550 20,-495 22,-450 25,-396 30,-330 33,-300 36,-275 44,-225 45,-220 50,-198 55,-180 60,-165 66,-150 75,-132 90,-110 99,-100$$

Calculate the sum for each pair.

$$1-9900=-9899 2-4950=-4948 3-3300=-3297 4-2475=-2471 5-1980=-1975 6-1650=-1644 9-1100=-1091 10-990=-980 11-900=-889 12-825=-813 15-660=-645 18-550=-532 20-495=-475 22-450=-428 25-396=-371 30-330=-300 33-300=-267 36-275=-239 44-225=-181 45-220=-175 50-198=-148 55-180=-125 60-165=-105 66-150=-84 75-132=-57 90-110=-20 99-100=-1$$

The solution is the pair that gives sum -1.

$$a=-100 b=99$$

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

$$\left(x-100\right)\left(x+99\right)$$

To find equation solutions, solve x-100=0 and x+99=0.

$$x=100 x=-99$$

Steps Using Factoring By Grouping

Calculate x to the power of 1 and get x.

$$x^{2}-x-9900=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-9900. To find a and b, set up a system to be solved.

$$a+b=-1 ab=1\left(-9900\right)=-9900$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -9900.

$$1,-9900 2,-4950 3,-3300 4,-2475 5,-1980 6,-1650 9,-1100 10,-990 11,-900 12,-825 15,-660 18,-550 20,-495 22,-450 25,-396 30,-330 33,-300 36,-275 44,-225 45,-220 50,-198 55,-180 60,-165 66,-150 75,-132 90,-110 99,-100$$

Calculate the sum for each pair.

$$1-9900=-9899 2-4950=-4948 3-3300=-3297 4-2475=-2471 5-1980=-1975 6-1650=-1644 9-1100=-1091 10-990=-980 11-900=-889 12-825=-813 15-660=-645 18-550=-532 20-495=-475 22-450=-428 25-396=-371 30-330=-300 33-300=-267 36-275=-239 44-225=-181 45-220=-175 50-198=-148 55-180=-125 60-165=-105 66-150=-84 75-132=-57 90-110=-20 99-100=-1$$

The solution is the pair that gives sum -1.

$$a=-100 b=99$$

Rewrite x^{2}-x-9900 as \left(x^{2}-100x\right)+\left(99x-9900\right).

$$\left(x^{2}-100x\right)+\left(99x-9900\right)$$

Factor out x in the first and 99 in the second group.

$$x\left(x-100\right)+99\left(x-100\right)$$

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

$$\left(x-100\right)\left(x+99\right)$$

To find equation solutions, solve x-100=0 and x+99=0.

$$x=100 x=-99$$

Steps Using the Quadratic Formula

Calculate x to the power of 1 and get x.

$$x^{2}-x-9900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -9900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-9900\right)}}{2}$$

Multiply -4 times -9900.

$$x=\frac{-\left(-1\right)±\sqrt{1+39600}}{2}$$

Add 1 to 39600.

$$x=\frac{-\left(-1\right)±\sqrt{39601}}{2}$$

Take the square root of 39601.

$$x=\frac{-\left(-1\right)±199}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±199}{2}$$

Now solve the equation x=\frac{1±199}{2} when ± is plus. Add 1 to 199.

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

Divide 200 by 2.

$$x=100$$

Now solve the equation x=\frac{1±199}{2} when ± is minus. Subtract 199 from 1.

$$x=-\frac{198}{2}$$

Divide -198 by 2.

$$x=-99$$

The equation is now solved.

$$x=100 x=-99$$

Steps for Completing the Square

Calculate x to the power of 1 and get x.

$$x^{2}-x-9900=0$$

Add 9900 to both sides. Anything plus zero gives itself.

$$x^{2}-x=9900$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=9900+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=9900+\frac{1}{4}$$

Add 9900 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{39601}{4}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{39601}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{39601}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{199}{2} x-\frac{1}{2}=-\frac{199}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=100 x=-99$$