Solve n^2+241n=9972

$$n ^ { 2 } + 241 n = 9972$$

Solve for n

$$n=-277$$

$$n=36$$

Steps Using Factoring

Subtract 9972 from both sides.

$$n^{2}+241n-9972=0$$

To solve the equation, factor n^{2}+241n-9972 using formula n^{2}+\left(a+b\right)n+ab=\left(n+a\right)\left(n+b\right). To find a and b, set up a system to be solved.

$$a+b=241 ab=-9972$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -9972.

$$-1,9972 -2,4986 -3,3324 -4,2493 -6,1662 -9,1108 -12,831 -18,554 -36,277$$

Calculate the sum for each pair.

$$-1+9972=9971 -2+4986=4984 -3+3324=3321 -4+2493=2489 -6+1662=1656 -9+1108=1099 -12+831=819 -18+554=536 -36+277=241$$

The solution is the pair that gives sum 241.

$$a=-36 b=277$$

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

$$\left(n-36\right)\left(n+277\right)$$

To find equation solutions, solve n-36=0 and n+277=0.

$$n=36 n=-277$$

Steps Using Factoring By Grouping

Subtract 9972 from both sides.

$$n^{2}+241n-9972=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as n^{2}+an+bn-9972. To find a and b, set up a system to be solved.

$$a+b=241 ab=1\left(-9972\right)=-9972$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -9972.

$$-1,9972 -2,4986 -3,3324 -4,2493 -6,1662 -9,1108 -12,831 -18,554 -36,277$$

Calculate the sum for each pair.

$$-1+9972=9971 -2+4986=4984 -3+3324=3321 -4+2493=2489 -6+1662=1656 -9+1108=1099 -12+831=819 -18+554=536 -36+277=241$$

The solution is the pair that gives sum 241.

$$a=-36 b=277$$

Rewrite n^{2}+241n-9972 as \left(n^{2}-36n\right)+\left(277n-9972\right).

$$\left(n^{2}-36n\right)+\left(277n-9972\right)$$

Factor out n in the first and 277 in the second group.

$$n\left(n-36\right)+277\left(n-36\right)$$

Factor out common term n-36 by using distributive property.

$$\left(n-36\right)\left(n+277\right)$$

To find equation solutions, solve n-36=0 and n+277=0.

$$n=36 n=-277$$

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.

$$n^{2}+241n=9972$$

Subtract 9972 from both sides of the equation.

$$n^{2}+241n-9972=9972-9972$$

Subtracting 9972 from itself leaves 0.

$$n^{2}+241n-9972=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 241 for b, and -9972 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-241±\sqrt{241^{2}-4\left(-9972\right)}}{2}$$

Square 241.

$$n=\frac{-241±\sqrt{58081-4\left(-9972\right)}}{2}$$

Multiply -4 times -9972.

$$n=\frac{-241±\sqrt{58081+39888}}{2}$$

Add 58081 to 39888.

$$n=\frac{-241±\sqrt{97969}}{2}$$

Take the square root of 97969.

$$n=\frac{-241±313}{2}$$

Now solve the equation n=\frac{-241±313}{2} when ± is plus. Add -241 to 313.

$$n=\frac{72}{2}$$

Divide 72 by 2.

$$n=36$$

Now solve the equation n=\frac{-241±313}{2} when ± is minus. Subtract 313 from -241.

$$n=-\frac{554}{2}$$

Divide -554 by 2.

$$n=-277$$

The equation is now solved.

$$n=36 n=-277$$

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.

$$n^{2}+241n=9972$$

Divide 241, the coefficient of the x term, by 2 to get \frac{241}{2}. Then add the square of \frac{241}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+241n+\left(\frac{241}{2}\right)^{2}=9972+\left(\frac{241}{2}\right)^{2}$$

Square \frac{241}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+241n+\frac{58081}{4}=9972+\frac{58081}{4}$$

Add 9972 to \frac{58081}{4}.

$$n^{2}+241n+\frac{58081}{4}=\frac{97969}{4}$$

Factor n^{2}+241n+\frac{58081}{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}.

$$\left(n+\frac{241}{2}\right)^{2}=\frac{97969}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{241}{2}\right)^{2}}=\sqrt{\frac{97969}{4}}$$

Simplify.

$$n+\frac{241}{2}=\frac{313}{2} n+\frac{241}{2}=-\frac{313}{2}$$

Subtract \frac{241}{2} from both sides of the equation.

$$n=36 n=-277$$