Solve 8b^2-72b+144

$$8 b ^ { 2 } - 72 b + 144$$

Factor

$$8\left(b-6\right)\left(b-3\right)$$

Solution Steps

Factor out 8.

$$8\left(b^{2}-9b+18\right)$$

Consider b^{2}-9b+18. Factor the expression by grouping. First, the expression needs to be rewritten as b^{2}+pb+qb+18. To find p and q, set up a system to be solved.

$$p+q=-9 pq=1\times 18=18$$

Since pq is positive, p and q have the same sign. Since p+q is negative, p and q are both negative. List all such integer pairs that give product 18.

$$-1,-18 -2,-9 -3,-6$$

Calculate the sum for each pair.

$$-1-18=-19 -2-9=-11 -3-6=-9$$

The solution is the pair that gives sum -9.

$$p=-6 q=-3$$

Rewrite b^{2}-9b+18 as \left(b^{2}-6b\right)+\left(-3b+18\right).

$$\left(b^{2}-6b\right)+\left(-3b+18\right)$$

Factor out b in the first and -3 in the second group.

$$b\left(b-6\right)-3\left(b-6\right)$$

Factor out common term b-6 by using distributive property.

$$\left(b-6\right)\left(b-3\right)$$

Rewrite the complete factored expression.

$$8\left(b-6\right)\left(b-3\right)$$

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.

$$8b^{2}-72b+144=0$$

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.

$$b=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 8\times 144}}{2\times 8}$$

Square -72.

$$b=\frac{-\left(-72\right)±\sqrt{5184-4\times 8\times 144}}{2\times 8}$$

Multiply -4 times 8.

$$b=\frac{-\left(-72\right)±\sqrt{5184-32\times 144}}{2\times 8}$$

Multiply -32 times 144.

$$b=\frac{-\left(-72\right)±\sqrt{5184-4608}}{2\times 8}$$

Add 5184 to -4608.

$$b=\frac{-\left(-72\right)±\sqrt{576}}{2\times 8}$$

Take the square root of 576.

$$b=\frac{-\left(-72\right)±24}{2\times 8}$$

The opposite of -72 is 72.

$$b=\frac{72±24}{2\times 8}$$

Multiply 2 times 8.

$$b=\frac{72±24}{16}$$

Now solve the equation b=\frac{72±24}{16} when ± is plus. Add 72 to 24.

$$b=\frac{96}{16}$$

Divide 96 by 16.

$$b=6$$

Now solve the equation b=\frac{72±24}{16} when ± is minus. Subtract 24 from 72.

$$b=\frac{48}{16}$$

Divide 48 by 16.

$$b=3$$

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 3 for x_{2}.

$$8b^{2}-72b+144=8\left(b-6\right)\left(b-3\right)$$

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 8

$$x ^ 2 -9x +18 = 0$$

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

$$r + s = 9 rs = 18$$

Two numbers r and s sum up to 9 exactly when the average of the two numbers is \frac{1}{2}*9 = \frac{9}{2}. 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>

$$r = \frac{9}{2} - u s = \frac{9}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 18

$$(\frac{9}{2} - u) (\frac{9}{2} + u) = 18$$

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

$$\frac{81}{4} - u^2 = 18$$

Simplify the expression by subtracting \frac{81}{4} on both sides

$$-u^2 = 18-\frac{81}{4} = -\frac{9}{4}$$

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

$$u^2 = \frac{9}{4} u = \pm\sqrt{\frac{9}{4}} = \pm \frac{3}{2}$$

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

$$r =\frac{9}{2} - \frac{3}{2} = 3 s = \frac{9}{2} + \frac{3}{2} = 6$$

Evaluate

$$8\left(b-6\right)\left(b-3\right)$$