Solve 2w^2+11w-63

$$2 w ^ { 2 } + 11 w - 63$$

Factor

$$\left(2w-7\right)\left(w+9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2w^{2}+aw+bw-63. To find a and b, set up a system to be solved.

$$a+b=11 ab=2\left(-63\right)=-126$$

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 -126.

$$-1,126 -2,63 -3,42 -6,21 -7,18 -9,14$$

Calculate the sum for each pair.

$$-1+126=125 -2+63=61 -3+42=39 -6+21=15 -7+18=11 -9+14=5$$

The solution is the pair that gives sum 11.

$$a=-7 b=18$$

Rewrite 2w^{2}+11w-63 as \left(2w^{2}-7w\right)+\left(18w-63\right).

$$\left(2w^{2}-7w\right)+\left(18w-63\right)$$

Factor out w in the first and 9 in the second group.

$$w\left(2w-7\right)+9\left(2w-7\right)$$

Factor out common term 2w-7 by using distributive property.

$$\left(2w-7\right)\left(w+9\right)$$

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.

$$2w^{2}+11w-63=0$$

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.

$$w=\frac{-11±\sqrt{11^{2}-4\times 2\left(-63\right)}}{2\times 2}$$

Square 11.

$$w=\frac{-11±\sqrt{121-4\times 2\left(-63\right)}}{2\times 2}$$

Multiply -4 times 2.

$$w=\frac{-11±\sqrt{121-8\left(-63\right)}}{2\times 2}$$

Multiply -8 times -63.

$$w=\frac{-11±\sqrt{121+504}}{2\times 2}$$

Add 121 to 504.

$$w=\frac{-11±\sqrt{625}}{2\times 2}$$

Take the square root of 625.

$$w=\frac{-11±25}{2\times 2}$$

Multiply 2 times 2.

$$w=\frac{-11±25}{4}$$

Now solve the equation w=\frac{-11±25}{4} when ± is plus. Add -11 to 25.

$$w=\frac{14}{4}$$

Reduce the fraction \frac{14}{4} to lowest terms by extracting and canceling out 2.

$$w=\frac{7}{2}$$

Now solve the equation w=\frac{-11±25}{4} when ± is minus. Subtract 25 from -11.

$$w=-\frac{36}{4}$$

Divide -36 by 4.

$$w=-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{2} for x_{1} and -9 for x_{2}.

$$2w^{2}+11w-63=2\left(w-\frac{7}{2}\right)\left(w-\left(-9\right)\right)$$

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

$$2w^{2}+11w-63=2\left(w-\frac{7}{2}\right)\left(w+9\right)$$

Subtract \frac{7}{2} from w by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$2w^{2}+11w-63=2\times \frac{2w-7}{2}\left(w+9\right)$$

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

$$2w^{2}+11w-63=\left(2w-7\right)\left(w+9\right)$$

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

$$x ^ 2 +\frac{11}{2}x -\frac{63}{2} = 0$$

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

$$r + s = -\frac{11}{2} rs = -\frac{63}{2}$$

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

$$r = -\frac{11}{4} - u s = -\frac{11}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{63}{2}

$$(-\frac{11}{4} - u) (-\frac{11}{4} + u) = -\frac{63}{2}$$

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

$$\frac{121}{16} - u^2 = -\frac{63}{2}$$

Simplify the expression by subtracting \frac{121}{16} on both sides

$$-u^2 = -\frac{63}{2}-\frac{121}{16} = -\frac{625}{16}$$

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

$$u^2 = \frac{625}{16} u = \pm\sqrt{\frac{625}{16}} = \pm \frac{25}{4}$$

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

$$r =-\frac{11}{4} - \frac{25}{4} = -9 s = -\frac{11}{4} + \frac{25}{4} = 3.500$$

Evaluate

$$\left(2w-7\right)\left(w+9\right)$$