Solve -x^2+40x-144

$$- x ^ { 2 } + 40 x - 144$$

Factor

$$\left(4-x\right)\left(x-36\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-144. To find a and b, set up a system to be solved.

$$a+b=40 ab=-\left(-144\right)=144$$

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 144.

$$1,144 2,72 3,48 4,36 6,24 8,18 9,16 12,12$$

Calculate the sum for each pair.

$$1+144=145 2+72=74 3+48=51 4+36=40 6+24=30 8+18=26 9+16=25 12+12=24$$

The solution is the pair that gives sum 40.

$$a=36 b=4$$

Rewrite -x^{2}+40x-144 as \left(-x^{2}+36x\right)+\left(4x-144\right).

$$\left(-x^{2}+36x\right)+\left(4x-144\right)$$

Factor out -x in the first and 4 in the second group.

$$-x\left(x-36\right)+4\left(x-36\right)$$

Factor out common term x-36 by using distributive property.

$$\left(x-36\right)\left(-x+4\right)$$

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.

$$-x^{2}+40x-144=0$$

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.

$$x=\frac{-40±\sqrt{40^{2}-4\left(-1\right)\left(-144\right)}}{2\left(-1\right)}$$

Square 40.

$$x=\frac{-40±\sqrt{1600-4\left(-1\right)\left(-144\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-40±\sqrt{1600+4\left(-144\right)}}{2\left(-1\right)}$$

Multiply 4 times -144.

$$x=\frac{-40±\sqrt{1600-576}}{2\left(-1\right)}$$

Add 1600 to -576.

$$x=\frac{-40±\sqrt{1024}}{2\left(-1\right)}$$

Take the square root of 1024.

$$x=\frac{-40±32}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-40±32}{-2}$$

Now solve the equation x=\frac{-40±32}{-2} when ± is plus. Add -40 to 32.

$$x=-\frac{8}{-2}$$

Divide -8 by -2.

$$x=4$$

Now solve the equation x=\frac{-40±32}{-2} when ± is minus. Subtract 32 from -40.

$$x=-\frac{72}{-2}$$

Divide -72 by -2.

$$x=36$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and 36 for x_{2}.

$$-x^{2}+40x-144=-\left(x-4\right)\left(x-36\right)$$

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.

$$x ^ 2 -40x +144 = 0$$

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

$$r + s = 40 rs = 144$$

Two numbers r and s sum up to 40 exactly when the average of the two numbers is \frac{1}{2}*40 = 20. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = 20 - u s = 20 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 144

$$(20 - u) (20 + u) = 144$$

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

$$400 - u^2 = 144$$

Simplify the expression by subtracting 400 on both sides

$$-u^2 = 144-400 = -256$$

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

$$u^2 = 256 u = \pm\sqrt{256} = \pm 16$$

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

$$r =20 - 16 = 4 s = 20 + 16 = 36$$

Evaluate

$$\left(4-x\right)\left(x-36\right)$$