Solve 2w^2+w-66

$$2 w ^ { 2 } + w - 66$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2w^{2}+aw+bw-66. To find a and b, set up a system to be solved.

$$a+b=1 ab=2\left(-66\right)=-132$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -132.

$$-1,132 -2,66 -3,44 -4,33 -6,22 -11,12$$

Calculate the sum for each pair.

$$-1+132=131 -2+66=64 -3+44=41 -4+33=29 -6+22=16 -11+12=1$$

The solution is the pair that gives sum 1.

$$a=-11 b=12$$

Rewrite 2w^{2}+w-66 as \left(2w^{2}-11w\right)+\left(12w-66\right).

$$\left(2w^{2}-11w\right)+\left(12w-66\right)$$

Factor out w in the first and 6 in the second group.

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

Factor out common term 2w-11 by using distributive property.

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

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.

$$2w^{2}+w-66=0$$

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.

$$w=\frac{-1±\sqrt{1^{2}-4\times 2\left(-66\right)}}{2\times 2}$$

Square 1.

$$w=\frac{-1±\sqrt{1-4\times 2\left(-66\right)}}{2\times 2}$$

Multiply -4 times 2.

$$w=\frac{-1±\sqrt{1-8\left(-66\right)}}{2\times 2}$$

Multiply -8 times -66.

$$w=\frac{-1±\sqrt{1+528}}{2\times 2}$$

Add 1 to 528.

$$w=\frac{-1±\sqrt{529}}{2\times 2}$$

Take the square root of 529.

$$w=\frac{-1±23}{2\times 2}$$

Multiply 2 times 2.

$$w=\frac{-1±23}{4}$$

Now solve the equation w=\frac{-1±23}{4} when ± is plus. Add -1 to 23.

$$w=\frac{22}{4}$$

Reduce the fraction \frac{22}{4} to lowest terms by extracting and canceling out 2.

$$w=\frac{11}{2}$$

Now solve the equation w=\frac{-1±23}{4} when ± is minus. Subtract 23 from -1.

$$w=-\frac{24}{4}$$

Divide -24 by 4.

$$w=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11}{2} for x_{1} and -6 for x_{2}.

$$2w^{2}+w-66=2\left(w-\frac{11}{2}\right)\left(w-\left(-6\right)\right)$$

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

$$2w^{2}+w-66=2\left(w-\frac{11}{2}\right)\left(w+6\right)$$

Subtract \frac{11}{2} from w by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$2w^{2}+w-66=2\times \frac{2w-11}{2}\left(w+6\right)$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2w^{2}+w-66=\left(2w-11\right)\left(w+6\right)$$

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 2

$$x ^ 2 +\frac{1}{2}x -33 = 0$$

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

$$r + s = -\frac{1}{2} rs = -33$$

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

$$r = -\frac{1}{4} - u s = -\frac{1}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -33

$$(-\frac{1}{4} - u) (-\frac{1}{4} + u) = -33$$

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

$$\frac{1}{16} - u^2 = -33$$

Simplify the expression by subtracting \frac{1}{16} on both sides

$$-u^2 = -33-\frac{1}{16} = -\frac{529}{16}$$

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

$$u^2 = \frac{529}{16} u = \pm\sqrt{\frac{529}{16}} = \pm \frac{23}{4}$$

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

$$r =-\frac{1}{4} - \frac{23}{4} = -6 s = -\frac{1}{4} + \frac{23}{4} = 5.500$$

Evaluate

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