Solve n^2-41n+400=0

$${ n }^{ 2 } -41n+400 = 0$$

Solve for n

$$n=16$$

$$n=25$$

Steps Using Factoring

To solve the equation, factor n^{2}-41n+400 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.

$$a+b=-41 ab=400$$

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 400.

$$-1,-400 -2,-200 -4,-100 -5,-80 -8,-50 -10,-40 -16,-25 -20,-20$$

Calculate the sum for each pair.

$$-1-400=-401 -2-200=-202 -4-100=-104 -5-80=-85 -8-50=-58 -10-40=-50 -16-25=-41 -20-20=-40$$

The solution is the pair that gives sum -41.

$$a=-25 b=-16$$

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

$$\left(n-25\right)\left(n-16\right)$$

To find equation solutions, solve n-25=0 and n-16=0.

$$n=25 n=16$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as n^{2}+an+bn+400. To find a and b, set up a system to be solved.

$$a+b=-41 ab=1\times 400=400$$

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 400.

$$-1,-400 -2,-200 -4,-100 -5,-80 -8,-50 -10,-40 -16,-25 -20,-20$$

Calculate the sum for each pair.

$$-1-400=-401 -2-200=-202 -4-100=-104 -5-80=-85 -8-50=-58 -10-40=-50 -16-25=-41 -20-20=-40$$

The solution is the pair that gives sum -41.

$$a=-25 b=-16$$

Rewrite n^{2}-41n+400 as \left(n^{2}-25n\right)+\left(-16n+400\right).

$$\left(n^{2}-25n\right)+\left(-16n+400\right)$$

Factor out n in the first and -16 in the second group.

$$n\left(n-25\right)-16\left(n-25\right)$$

Factor out common term n-25 by using distributive property.

$$\left(n-25\right)\left(n-16\right)$$

To find equation solutions, solve n-25=0 and n-16=0.

$$n=25 n=16$$

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.

$$n^{2}-41n+400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -41 for b, and 400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-41\right)±\sqrt{\left(-41\right)^{2}-4\times 400}}{2}$$

Square -41.

$$n=\frac{-\left(-41\right)±\sqrt{1681-4\times 400}}{2}$$

Multiply -4 times 400.

$$n=\frac{-\left(-41\right)±\sqrt{1681-1600}}{2}$$

Add 1681 to -1600.

$$n=\frac{-\left(-41\right)±\sqrt{81}}{2}$$

Take the square root of 81.

$$n=\frac{-\left(-41\right)±9}{2}$$

The opposite of -41 is 41.

$$n=\frac{41±9}{2}$$

Now solve the equation n=\frac{41±9}{2} when ± is plus. Add 41 to 9.

$$n=\frac{50}{2}$$

Divide 50 by 2.

$$n=25$$

Now solve the equation n=\frac{41±9}{2} when ± is minus. Subtract 9 from 41.

$$n=\frac{32}{2}$$

Divide 32 by 2.

$$n=16$$

The equation is now solved.

$$n=25 n=16$$

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.

$$n^{2}-41n+400=0$$

Subtract 400 from both sides of the equation.

$$n^{2}-41n+400-400=-400$$

Subtracting 400 from itself leaves 0.

$$n^{2}-41n=-400$$

Divide -41, the coefficient of the x term, by 2 to get -\frac{41}{2}. Then add the square of -\frac{41}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-41n+\left(-\frac{41}{2}\right)^{2}=-400+\left(-\frac{41}{2}\right)^{2}$$

Square -\frac{41}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-41n+\frac{1681}{4}=-400+\frac{1681}{4}$$

Add -400 to \frac{1681}{4}.

$$n^{2}-41n+\frac{1681}{4}=\frac{81}{4}$$

Factor n^{2}-41n+\frac{1681}{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}.

$$\left(n-\frac{41}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{41}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$n-\frac{41}{2}=\frac{9}{2} n-\frac{41}{2}=-\frac{9}{2}$$

Add \frac{41}{2} to both sides of the equation.

$$n=25 n=16$$