Solve x^2-x=1980

$${ x }^{ 2 } -x=1980$$

Solve for x

$$x=-44$$

$$x=45$$

Steps Using Factoring

Subtract 1980 from both sides.

$$x^{2}-x-1980=0$$

To solve the equation, factor x^{2}-x-1980 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.

$$a+b=-1 ab=-1980$$

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 -1980.

$$1,-1980 2,-990 3,-660 4,-495 5,-396 6,-330 9,-220 10,-198 11,-180 12,-165 15,-132 18,-110 20,-99 22,-90 30,-66 33,-60 36,-55 44,-45$$

Calculate the sum for each pair.

$$1-1980=-1979 2-990=-988 3-660=-657 4-495=-491 5-396=-391 6-330=-324 9-220=-211 10-198=-188 11-180=-169 12-165=-153 15-132=-117 18-110=-92 20-99=-79 22-90=-68 30-66=-36 33-60=-27 36-55=-19 44-45=-1$$

The solution is the pair that gives sum -1.

$$a=-45 b=44$$

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

$$\left(x-45\right)\left(x+44\right)$$

To find equation solutions, solve x-45=0 and x+44=0.

$$x=45 x=-44$$

Steps Using Factoring By Grouping

Subtract 1980 from both sides.

$$x^{2}-x-1980=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-1980. To find a and b, set up a system to be solved.

$$a+b=-1 ab=1\left(-1980\right)=-1980$$

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 -1980.

$$1,-1980 2,-990 3,-660 4,-495 5,-396 6,-330 9,-220 10,-198 11,-180 12,-165 15,-132 18,-110 20,-99 22,-90 30,-66 33,-60 36,-55 44,-45$$

Calculate the sum for each pair.

$$1-1980=-1979 2-990=-988 3-660=-657 4-495=-491 5-396=-391 6-330=-324 9-220=-211 10-198=-188 11-180=-169 12-165=-153 15-132=-117 18-110=-92 20-99=-79 22-90=-68 30-66=-36 33-60=-27 36-55=-19 44-45=-1$$

The solution is the pair that gives sum -1.

$$a=-45 b=44$$

Rewrite x^{2}-x-1980 as \left(x^{2}-45x\right)+\left(44x-1980\right).

$$\left(x^{2}-45x\right)+\left(44x-1980\right)$$

Factor out x in the first and 44 in the second group.

$$x\left(x-45\right)+44\left(x-45\right)$$

Factor out common term x-45 by using distributive property.

$$\left(x-45\right)\left(x+44\right)$$

To find equation solutions, solve x-45=0 and x+44=0.

$$x=45 x=-44$$

Steps Using the Quadratic Formula

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.

$$x^{2}-x=1980$$

Subtract 1980 from both sides of the equation.

$$x^{2}-x-1980=1980-1980$$

Subtracting 1980 from itself leaves 0.

$$x^{2}-x-1980=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -1980 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1980\right)}}{2}$$

Multiply -4 times -1980.

$$x=\frac{-\left(-1\right)±\sqrt{1+7920}}{2}$$

Add 1 to 7920.

$$x=\frac{-\left(-1\right)±\sqrt{7921}}{2}$$

Take the square root of 7921.

$$x=\frac{-\left(-1\right)±89}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±89}{2}$$

Now solve the equation x=\frac{1±89}{2} when ± is plus. Add 1 to 89.

$$x=\frac{90}{2}$$

Divide 90 by 2.

$$x=45$$

Now solve the equation x=\frac{1±89}{2} when ± is minus. Subtract 89 from 1.

$$x=-\frac{88}{2}$$

Divide -88 by 2.

$$x=-44$$

The equation is now solved.

$$x=45 x=-44$$

Steps for Completing the Square

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.

$$x^{2}-x=1980$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=1980+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=1980+\frac{1}{4}$$

Add 1980 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{7921}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{7921}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{7921}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{89}{2} x-\frac{1}{2}=-\frac{89}{2}$$

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

$$x=45 x=-44$$