Solve f(x)=-x^2+24x+180

$$f ( x ) = - x ^ { 2 } + 24 x + 180$$

Factor

$$-\left(x-30\right)\left(x+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+180. To find a and b, set up a system to be solved.

$$a+b=24 ab=-180=-180$$

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 -180.

$$-1,180 -2,90 -3,60 -4,45 -5,36 -6,30 -9,20 -10,18 -12,15$$

Calculate the sum for each pair.

$$-1+180=179 -2+90=88 -3+60=57 -4+45=41 -5+36=31 -6+30=24 -9+20=11 -10+18=8 -12+15=3$$

The solution is the pair that gives sum 24.

$$a=30 b=-6$$

Rewrite -x^{2}+24x+180 as \left(-x^{2}+30x\right)+\left(-6x+180\right).

$$\left(-x^{2}+30x\right)+\left(-6x+180\right)$$

Factor out -x in the first and -6 in the second group.

$$-x\left(x-30\right)-6\left(x-30\right)$$

Factor out common term x-30 by using distributive property.

$$\left(x-30\right)\left(-x-6\right)$$

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.

$$-x^{2}+24x+180=0$$

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.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-1\right)\times 180}}{2\left(-1\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-1\right)\times 180}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-24±\sqrt{576+4\times 180}}{2\left(-1\right)}$$

Multiply 4 times 180.

$$x=\frac{-24±\sqrt{576+720}}{2\left(-1\right)}$$

Add 576 to 720.

$$x=\frac{-24±\sqrt{1296}}{2\left(-1\right)}$$

Take the square root of 1296.

$$x=\frac{-24±36}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-24±36}{-2}$$

Now solve the equation x=\frac{-24±36}{-2} when ± is plus. Add -24 to 36.

$$x=\frac{12}{-2}$$

Divide 12 by -2.

$$x=-6$$

Now solve the equation x=\frac{-24±36}{-2} when ± is minus. Subtract 36 from -24.

$$x=-\frac{60}{-2}$$

Divide -60 by -2.

$$x=30$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6 for x_{1} and 30 for x_{2}.

$$-x^{2}+24x+180=-\left(x-\left(-6\right)\right)\left(x-30\right)$$

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

$$-x^{2}+24x+180=-\left(x+6\right)\left(x-30\right)$$

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.

$$x ^ 2 -24x -180 = 0$$

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

$$r + s = 24 rs = -180$$

Two numbers r and s sum up to 24 exactly when the average of the two numbers is \frac{1}{2}*24 = 12. 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>

$$r = 12 - u s = 12 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -180

$$(12 - u) (12 + u) = -180$$

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

$$144 - u^2 = -180$$

Simplify the expression by subtracting 144 on both sides

$$-u^2 = -180-144 = -324$$

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

$$u^2 = 324 u = \pm\sqrt{324} = \pm 18$$

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

$$r =12 - 18 = -6 s = 12 + 18 = 30$$

Evaluate

$$-\left(x-30\right)\left(x+6\right)$$