Solve n^2-+43n+432

$$n ^ { 2 } - + 43 n + 432$$

Factor

$$\left(n-27\right)\left(n-16\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as n^{2}+an+bn+432. To find a and b, set up a system to be solved.

$$a+b=-43 ab=1\times 432=432$$

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 432.

$$-1,-432 -2,-216 -3,-144 -4,-108 -6,-72 -8,-54 -9,-48 -12,-36 -16,-27 -18,-24$$

Calculate the sum for each pair.

$$-1-432=-433 -2-216=-218 -3-144=-147 -4-108=-112 -6-72=-78 -8-54=-62 -9-48=-57 -12-36=-48 -16-27=-43 -18-24=-42$$

The solution is the pair that gives sum -43.

$$a=-27 b=-16$$

Rewrite n^{2}-43n+432 as \left(n^{2}-27n\right)+\left(-16n+432\right).

$$\left(n^{2}-27n\right)+\left(-16n+432\right)$$

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

$$n\left(n-27\right)-16\left(n-27\right)$$

Factor out common term n-27 by using distributive property.

$$\left(n-27\right)\left(n-16\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$n^{2}-43n+432=0$$

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.

$$n=\frac{-\left(-43\right)±\sqrt{\left(-43\right)^{2}-4\times 432}}{2}$$

Square -43.

$$n=\frac{-\left(-43\right)±\sqrt{1849-4\times 432}}{2}$$

Multiply -4 times 432.

$$n=\frac{-\left(-43\right)±\sqrt{1849-1728}}{2}$$

Add 1849 to -1728.

$$n=\frac{-\left(-43\right)±\sqrt{121}}{2}$$

Take the square root of 121.

$$n=\frac{-\left(-43\right)±11}{2}$$

The opposite of -43 is 43.

$$n=\frac{43±11}{2}$$

Now solve the equation n=\frac{43±11}{2} when ± is plus. Add 43 to 11.

$$n=\frac{54}{2}$$

Divide 54 by 2.

$$n=27$$

Now solve the equation n=\frac{43±11}{2} when ± is minus. Subtract 11 from 43.

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

Divide 32 by 2.

$$n=16$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 27 for x_{1} and 16 for x_{2}.

$$n^{2}-43n+432=\left(n-27\right)\left(n-16\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 -43x +432 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = 43 rs = 432$$

Two numbers r and s sum up to 43 exactly when the average of the two numbers is \frac{1}{2}*43 = \frac{43}{2}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{43}{2} - u s = \frac{43}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 432

$$(\frac{43}{2} - u) (\frac{43}{2} + u) = 432$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{1849}{4} - u^2 = 432$$

Simplify the expression by subtracting \frac{1849}{4} on both sides

$$-u^2 = 432-\frac{1849}{4} = -\frac{121}{4}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{121}{4} u = \pm\sqrt{\frac{121}{4}} = \pm \frac{11}{2}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{43}{2} - \frac{11}{2} = 16 s = \frac{43}{2} + \frac{11}{2} = 27$$

Evaluate

$$\left(n-27\right)\left(n-16\right)$$