Solve 3b^2-72b+144

$$3 b ^ { 2 } - 72 b + 144$$

Factor

$$3\left(b-\left(12-4\sqrt{6}\right)\right)\left(b-\left(4\sqrt{6}+12\right)\right)$$

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.

$$3b^{2}-72b+144=0$$

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.

$$b=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 3\times 144}}{2\times 3}$$

Square -72.

$$b=\frac{-\left(-72\right)±\sqrt{5184-4\times 3\times 144}}{2\times 3}$$

Multiply -4 times 3.

$$b=\frac{-\left(-72\right)±\sqrt{5184-12\times 144}}{2\times 3}$$

Multiply -12 times 144.

$$b=\frac{-\left(-72\right)±\sqrt{5184-1728}}{2\times 3}$$

Add 5184 to -1728.

$$b=\frac{-\left(-72\right)±\sqrt{3456}}{2\times 3}$$

Take the square root of 3456.

$$b=\frac{-\left(-72\right)±24\sqrt{6}}{2\times 3}$$

The opposite of -72 is 72.

$$b=\frac{72±24\sqrt{6}}{2\times 3}$$

Multiply 2 times 3.

$$b=\frac{72±24\sqrt{6}}{6}$$

Now solve the equation b=\frac{72±24\sqrt{6}}{6} when ± is plus. Add 72 to 24\sqrt{6}.

$$b=\frac{24\sqrt{6}+72}{6}$$

Divide 72+24\sqrt{6} by 6.

$$b=4\sqrt{6}+12$$

Now solve the equation b=\frac{72±24\sqrt{6}}{6} when ± is minus. Subtract 24\sqrt{6} from 72.

$$b=\frac{72-24\sqrt{6}}{6}$$

Divide 72-24\sqrt{6} by 6.

$$b=12-4\sqrt{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12+4\sqrt{6} for x_{1} and 12-4\sqrt{6} for x_{2}.

$$3b^{2}-72b+144=3\left(b-\left(4\sqrt{6}+12\right)\right)\left(b-\left(12-4\sqrt{6}\right)\right)$$

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.This is achieved by dividing both sides of the equation by 3

$$x ^ 2 -24x +48 = 0$$

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

$$r + s = 24 rs = 48$$

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 = 48

$$(12 - u) (12 + u) = 48$$

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

$$144 - u^2 = 48$$

Simplify the expression by subtracting 144 on both sides

$$-u^2 = 48-144 = -96$$

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

$$u^2 = 96 u = \pm\sqrt{96} = \pm \sqrt{96}$$

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

$$r =12 - \sqrt{96} = 2.202 s = 12 + \sqrt{96} = 21.798$$

Evaluate

$$3\left(b^{2}-24b+48\right)$$