Solve 8w^2-80w+72=

$$8 w ^ { 2 } - 80 w + 72 =$$

Factor

$$8\left(w-9\right)\left(w-1\right)$$

Solution Steps

Factor out 8.

$$8\left(w^{2}-10w+9\right)$$

Consider w^{2}-10w+9. Factor the expression by grouping. First, the expression needs to be rewritten as w^{2}+aw+bw+9. To find a and b, set up a system to be solved.

$$a+b=-10 ab=1\times 9=9$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 9.

$$-1,-9 -3,-3$$

Calculate the sum for each pair.

$$-1-9=-10 -3-3=-6$$

The solution is the pair that gives sum -10.

$$a=-9 b=-1$$

Rewrite w^{2}-10w+9 as \left(w^{2}-9w\right)+\left(-w+9\right).

$$\left(w^{2}-9w\right)+\left(-w+9\right)$$

Factor out w in the first and -1 in the second group.

$$w\left(w-9\right)-\left(w-9\right)$$

Factor out common term w-9 by using distributive property.

$$\left(w-9\right)\left(w-1\right)$$

Rewrite the complete factored expression.

$$8\left(w-9\right)\left(w-1\right)$$

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.

$$8w^{2}-80w+72=0$$

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.

$$w=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 8\times 72}}{2\times 8}$$

Square -80.

$$w=\frac{-\left(-80\right)±\sqrt{6400-4\times 8\times 72}}{2\times 8}$$

Multiply -4 times 8.

$$w=\frac{-\left(-80\right)±\sqrt{6400-32\times 72}}{2\times 8}$$

Multiply -32 times 72.

$$w=\frac{-\left(-80\right)±\sqrt{6400-2304}}{2\times 8}$$

Add 6400 to -2304.

$$w=\frac{-\left(-80\right)±\sqrt{4096}}{2\times 8}$$

Take the square root of 4096.

$$w=\frac{-\left(-80\right)±64}{2\times 8}$$

The opposite of -80 is 80.

$$w=\frac{80±64}{2\times 8}$$

Multiply 2 times 8.

$$w=\frac{80±64}{16}$$

Now solve the equation w=\frac{80±64}{16} when ± is plus. Add 80 to 64.

$$w=\frac{144}{16}$$

Divide 144 by 16.

$$w=9$$

Now solve the equation w=\frac{80±64}{16} when ± is minus. Subtract 64 from 80.

$$w=\frac{16}{16}$$

Divide 16 by 16.

$$w=1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and 1 for x_{2}.

$$8w^{2}-80w+72=8\left(w-9\right)\left(w-1\right)$$

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

$$x ^ 2 -10x +9 = 0$$

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

$$r + s = 10 rs = 9$$

Two numbers r and s sum up to 10 exactly when the average of the two numbers is \frac{1}{2}*10 = 5. 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>

$$r = 5 - u s = 5 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 9

$$(5 - u) (5 + u) = 9$$

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

$$25 - u^2 = 9$$

Simplify the expression by subtracting 25 on both sides

$$-u^2 = 9-25 = -16$$

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

$$u^2 = 16 u = \pm\sqrt{16} = \pm 4$$

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

$$r =5 - 4 = 1 s = 5 + 4 = 9$$

Evaluate

$$8\left(w-9\right)\left(w-1\right)$$