Solve 6w^2-66w-72

$$6 w ^ { 2 } - 66 w - 72$$

Factor

$$6\left(w-12\right)\left(w+1\right)$$

Solution Steps

Factor out 6.

$$6\left(w^{2}-11w-12\right)$$

Consider w^{2}-11w-12. Factor the expression by grouping. First, the expression needs to be rewritten as w^{2}+aw+bw-12. To find a and b, set up a system to be solved.

$$a+b=-11 ab=1\left(-12\right)=-12$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -12.

$$1,-12 2,-6 3,-4$$

Calculate the sum for each pair.

$$1-12=-11 2-6=-4 3-4=-1$$

The solution is the pair that gives sum -11.

$$a=-12 b=1$$

Rewrite w^{2}-11w-12 as \left(w^{2}-12w\right)+\left(w-12\right).

$$\left(w^{2}-12w\right)+\left(w-12\right)$$

Factor out w in w^{2}-12w.

$$w\left(w-12\right)+w-12$$

Factor out common term w-12 by using distributive property.

$$\left(w-12\right)\left(w+1\right)$$

Rewrite the complete factored expression.

$$6\left(w-12\right)\left(w+1\right)$$

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.

$$6w^{2}-66w-72=0$$

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.

$$w=\frac{-\left(-66\right)±\sqrt{\left(-66\right)^{2}-4\times 6\left(-72\right)}}{2\times 6}$$

Square -66.

$$w=\frac{-\left(-66\right)±\sqrt{4356-4\times 6\left(-72\right)}}{2\times 6}$$

Multiply -4 times 6.

$$w=\frac{-\left(-66\right)±\sqrt{4356-24\left(-72\right)}}{2\times 6}$$

Multiply -24 times -72.

$$w=\frac{-\left(-66\right)±\sqrt{4356+1728}}{2\times 6}$$

Add 4356 to 1728.

$$w=\frac{-\left(-66\right)±\sqrt{6084}}{2\times 6}$$

Take the square root of 6084.

$$w=\frac{-\left(-66\right)±78}{2\times 6}$$

The opposite of -66 is 66.

$$w=\frac{66±78}{2\times 6}$$

Multiply 2 times 6.

$$w=\frac{66±78}{12}$$

Now solve the equation w=\frac{66±78}{12} when ± is plus. Add 66 to 78.

$$w=\frac{144}{12}$$

Divide 144 by 12.

$$w=12$$

Now solve the equation w=\frac{66±78}{12} when ± is minus. Subtract 78 from 66.

$$w=-\frac{12}{12}$$

Divide -12 by 12.

$$w=-1$$

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

$$6w^{2}-66w-72=6\left(w-12\right)\left(w-\left(-1\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$6w^{2}-66w-72=6\left(w-12\right)\left(w+1\right)$$

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 6

$$x ^ 2 -11x -12 = 0$$

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

$$r + s = 11 rs = -12$$

Two numbers r and s sum up to 11 exactly when the average of the two numbers is \frac{1}{2}*11 = \frac{11}{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>

$$r = \frac{11}{2} - u s = \frac{11}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -12

$$(\frac{11}{2} - u) (\frac{11}{2} + u) = -12$$

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

$$\frac{121}{4} - u^2 = -12$$

Simplify the expression by subtracting \frac{121}{4} on both sides

$$-u^2 = -12-\frac{121}{4} = -\frac{169}{4}$$

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

$$u^2 = \frac{169}{4} u = \pm\sqrt{\frac{169}{4}} = \pm \frac{13}{2}$$

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

$$r =\frac{11}{2} - \frac{13}{2} = -1 s = \frac{11}{2} + \frac{13}{2} = 12$$

Evaluate

$$6\left(w-12\right)\left(w+1\right)$$