Solve x^2-32x+72

$$x ^ { 2 } - 32 x + 72$$

Factor

$$\left(x-\left(16-2\sqrt{46}\right)\right)\left(x-\left(2\sqrt{46}+16\right)\right)$$

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.

$$x^{2}-32x+72=0$$

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.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 72}}{2}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 72}}{2}$$

Multiply -4 times 72.

$$x=\frac{-\left(-32\right)±\sqrt{1024-288}}{2}$$

Add 1024 to -288.

$$x=\frac{-\left(-32\right)±\sqrt{736}}{2}$$

Take the square root of 736.

$$x=\frac{-\left(-32\right)±4\sqrt{46}}{2}$$

The opposite of -32 is 32.

$$x=\frac{32±4\sqrt{46}}{2}$$

Now solve the equation x=\frac{32±4\sqrt{46}}{2} when ± is plus. Add 32 to 4\sqrt{46}.

$$x=\frac{4\sqrt{46}+32}{2}$$

Divide 32+4\sqrt{46} by 2.

$$x=2\sqrt{46}+16$$

Now solve the equation x=\frac{32±4\sqrt{46}}{2} when ± is minus. Subtract 4\sqrt{46} from 32.

$$x=\frac{32-4\sqrt{46}}{2}$$

Divide 32-4\sqrt{46} by 2.

$$x=16-2\sqrt{46}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 16+2\sqrt{46} for x_{1} and 16-2\sqrt{46} for x_{2}.

$$x^{2}-32x+72=\left(x-\left(2\sqrt{46}+16\right)\right)\left(x-\left(16-2\sqrt{46}\right)\right)$$

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.

$$x ^ 2 -32x +72 = 0$$

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

$$r + s = 32 rs = 72$$

Two numbers r and s sum up to 32 exactly when the average of the two numbers is \frac{1}{2}*32 = 16. 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>

$$r = 16 - u s = 16 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 72

$$(16 - u) (16 + u) = 72$$

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

$$256 - u^2 = 72$$

Simplify the expression by subtracting 256 on both sides

$$-u^2 = 72-256 = -184$$

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

$$u^2 = 184 u = \pm\sqrt{184} = \pm \sqrt{184}$$

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

$$r =16 - \sqrt{184} = 2.435 s = 16 + \sqrt{184} = 29.565$$

Evaluate

$$x^{2}-32x+72$$