Solve -x^2+36x-180

$$- x ^ { 2 } + 36 x - 180$$

Factor

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

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.

$$a+b=36 ab=-\left(-180\right)=180$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. 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.

$$1+180=181 2+90=92 3+60=63 4+45=49 5+36=41 6+30=36 9+20=29 10+18=28 12+15=27$$

The solution is the pair that gives sum 36.

$$a=30 b=6$$

Rewrite -x^{2}+36x-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.

$$-x^{2}+36x-180=0$$

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.

$$x=\frac{-36±\sqrt{36^{2}-4\left(-1\right)\left(-180\right)}}{2\left(-1\right)}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\left(-1\right)\left(-180\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-36±\sqrt{1296+4\left(-180\right)}}{2\left(-1\right)}$$

Multiply 4 times -180.

$$x=\frac{-36±\sqrt{1296-720}}{2\left(-1\right)}$$

Add 1296 to -720.

$$x=\frac{-36±\sqrt{576}}{2\left(-1\right)}$$

Take the square root of 576.

$$x=\frac{-36±24}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-36±24}{-2}$$

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

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

Divide -12 by -2.

$$x=6$$

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

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

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}.

$$-x^{2}+36x-180=-\left(x-6\right)\left(x-30\right)$$

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.

$$x ^ 2 -36x +180 = 0$$

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

$$r + s = 36 rs = 180$$

Two numbers r and s sum up to 36 exactly when the average of the two numbers is \frac{1}{2}*36 = 18. 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>

$$r = 18 - u s = 18 + u$$

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

$$(18 - u) (18 + u) = 180$$

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

$$324 - u^2 = 180$$

Simplify the expression by subtracting 324 on both sides

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

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

$$u^2 = 144 u = \pm\sqrt{144} = \pm 12$$

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

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

Evaluate

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