Solve 4n^2+16n-180

$$4 n ^ { 2 } + 16 n - 180$$

Factor

$$4\left(n-5\right)\left(n+9\right)$$

Solution Steps

Factor out 4.

$$4\left(n^{2}+4n-45\right)$$

Consider n^{2}+4n-45. Factor the expression by grouping. First, the expression needs to be rewritten as n^{2}+an+bn-45. To find a and b, set up a system to be solved.

$$a+b=4 ab=1\left(-45\right)=-45$$

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 -45.

$$-1,45 -3,15 -5,9$$

Calculate the sum for each pair.

$$-1+45=44 -3+15=12 -5+9=4$$

The solution is the pair that gives sum 4.

$$a=-5 b=9$$

Rewrite n^{2}+4n-45 as \left(n^{2}-5n\right)+\left(9n-45\right).

$$\left(n^{2}-5n\right)+\left(9n-45\right)$$

Factor out n in the first and 9 in the second group.

$$n\left(n-5\right)+9\left(n-5\right)$$

Factor out common term n-5 by using distributive property.

$$\left(n-5\right)\left(n+9\right)$$

Rewrite the complete factored expression.

$$4\left(n-5\right)\left(n+9\right)$$

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.

$$4n^{2}+16n-180=0$$

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.

$$n=\frac{-16±\sqrt{16^{2}-4\times 4\left(-180\right)}}{2\times 4}$$

Square 16.

$$n=\frac{-16±\sqrt{256-4\times 4\left(-180\right)}}{2\times 4}$$

Multiply -4 times 4.

$$n=\frac{-16±\sqrt{256-16\left(-180\right)}}{2\times 4}$$

Multiply -16 times -180.

$$n=\frac{-16±\sqrt{256+2880}}{2\times 4}$$

Add 256 to 2880.

$$n=\frac{-16±\sqrt{3136}}{2\times 4}$$

Take the square root of 3136.

$$n=\frac{-16±56}{2\times 4}$$

Multiply 2 times 4.

$$n=\frac{-16±56}{8}$$

Now solve the equation n=\frac{-16±56}{8} when ± is plus. Add -16 to 56.

$$n=\frac{40}{8}$$

Divide 40 by 8.

$$n=5$$

Now solve the equation n=\frac{-16±56}{8} when ± is minus. Subtract 56 from -16.

$$n=-\frac{72}{8}$$

Divide -72 by 8.

$$n=-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and -9 for x_{2}.

$$4n^{2}+16n-180=4\left(n-5\right)\left(n-\left(-9\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$4n^{2}+16n-180=4\left(n-5\right)\left(n+9\right)$$

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.This is achieved by dividing both sides of the equation by 4

$$x ^ 2 +4x -45 = 0$$

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

$$r + s = -4 rs = -45$$

Two numbers r and s sum up to -4 exactly when the average of the two numbers is \frac{1}{2}*-4 = -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>

$$r = -2 - u s = -2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -45

$$(-2 - u) (-2 + u) = -45$$

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

$$4 - u^2 = -45$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -45-4 = -49$$

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

$$u^2 = 49 u = \pm\sqrt{49} = \pm 7$$

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

$$r =-2 - 7 = -9 s = -2 + 7 = 5$$

Evaluate

$$4\left(n-5\right)\left(n+9\right)$$