Solve 11w^2-44w-132

$$11 w ^ { 2 } - 44 w - 132$$

Factor

$$11\left(w-6\right)\left(w+2\right)$$

Solution Steps

Factor out 11.

$$11\left(w^{2}-4w-12\right)$$

Consider w^{2}-4w-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.

$$a+b=-4 ab=1\left(-12\right)=-12$$

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 -4.

$$a=-6 b=2$$

Rewrite w^{2}-4w-12 as \left(w^{2}-6w\right)+\left(2w-12\right).

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

Factor out w in the first and 2 in the second group.

$$w\left(w-6\right)+2\left(w-6\right)$$

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

$$\left(w-6\right)\left(w+2\right)$$

Rewrite the complete factored expression.

$$11\left(w-6\right)\left(w+2\right)$$

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.

$$11w^{2}-44w-132=0$$

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.

$$w=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\times 11\left(-132\right)}}{2\times 11}$$

Square -44.

$$w=\frac{-\left(-44\right)±\sqrt{1936-4\times 11\left(-132\right)}}{2\times 11}$$

Multiply -4 times 11.

$$w=\frac{-\left(-44\right)±\sqrt{1936-44\left(-132\right)}}{2\times 11}$$

Multiply -44 times -132.

$$w=\frac{-\left(-44\right)±\sqrt{1936+5808}}{2\times 11}$$

Add 1936 to 5808.

$$w=\frac{-\left(-44\right)±\sqrt{7744}}{2\times 11}$$

Take the square root of 7744.

$$w=\frac{-\left(-44\right)±88}{2\times 11}$$

The opposite of -44 is 44.

$$w=\frac{44±88}{2\times 11}$$

Multiply 2 times 11.

$$w=\frac{44±88}{22}$$

Now solve the equation w=\frac{44±88}{22} when ± is plus. Add 44 to 88.

$$w=\frac{132}{22}$$

Divide 132 by 22.

$$w=6$$

Now solve the equation w=\frac{44±88}{22} when ± is minus. Subtract 88 from 44.

$$w=-\frac{44}{22}$$

Divide -44 by 22.

$$w=-2$$

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 -2 for x_{2}.

$$11w^{2}-44w-132=11\left(w-6\right)\left(w-\left(-2\right)\right)$$

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

$$11w^{2}-44w-132=11\left(w-6\right)\left(w+2\right)$$

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 11

$$x ^ 2 -4x -12 = 0$$

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

$$r + s = 4 rs = -12$$

Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 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>

$$r = 2 - u s = 2 + u$$

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

$$(2 - u) (2 + u) = -12$$

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

$$4 - u^2 = -12$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -12-4 = -16$$

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.

$$r =2 - 4 = -2 s = 2 + 4 = 6$$

Evaluate

$$11\left(w-6\right)\left(w+2\right)$$