Solve 2w^2+5w-63

$$2 w ^ { 2 } + 5 w - 63$$

Factor

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

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.

$$a+b=5 ab=2\left(-63\right)=-126$$

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 5.

$$a=-9 b=14$$

Rewrite 2w^{2}+5w-63 as \left(2w^{2}-9w\right)+\left(14w-63\right).

$$\left(2w^{2}-9w\right)+\left(14w-63\right)$$

Factor out w in the first and 7 in the second group.

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

Factor out common term 2w-9 by using distributive property.

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

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.

$$2w^{2}+5w-63=0$$

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.

$$w=\frac{-5±\sqrt{5^{2}-4\times 2\left(-63\right)}}{2\times 2}$$

Square 5.

$$w=\frac{-5±\sqrt{25-4\times 2\left(-63\right)}}{2\times 2}$$

Multiply -4 times 2.

$$w=\frac{-5±\sqrt{25-8\left(-63\right)}}{2\times 2}$$

Multiply -8 times -63.

$$w=\frac{-5±\sqrt{25+504}}{2\times 2}$$

Add 25 to 504.

$$w=\frac{-5±\sqrt{529}}{2\times 2}$$

Take the square root of 529.

$$w=\frac{-5±23}{2\times 2}$$

Multiply 2 times 2.

$$w=\frac{-5±23}{4}$$

Now solve the equation w=\frac{-5±23}{4} when ± is plus. Add -5 to 23.

$$w=\frac{18}{4}$$

Reduce the fraction \frac{18}{4} to lowest terms by extracting and canceling out 2.

$$w=\frac{9}{2}$$

Now solve the equation w=\frac{-5±23}{4} when ± is minus. Subtract 23 from -5.

$$w=-\frac{28}{4}$$

Divide -28 by 4.

$$w=-7$$

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

$$2w^{2}+5w-63=2\left(w-\frac{9}{2}\right)\left(w-\left(-7\right)\right)$$

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

$$2w^{2}+5w-63=2\left(w-\frac{9}{2}\right)\left(w+7\right)$$

Subtract \frac{9}{2} from w by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$2w^{2}+5w-63=2\times \frac{2w-9}{2}\left(w+7\right)$$

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

$$2w^{2}+5w-63=\left(2w-9\right)\left(w+7\right)$$

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

$$x ^ 2 +\frac{5}{2}x -\frac{63}{2} = 0$$

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

$$r + s = -\frac{5}{2} rs = -\frac{63}{2}$$

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

$$r = -\frac{5}{4} - u s = -\frac{5}{4} + u$$

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

$$(-\frac{5}{4} - u) (-\frac{5}{4} + u) = -\frac{63}{2}$$

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

$$\frac{25}{16} - u^2 = -\frac{63}{2}$$

Simplify the expression by subtracting \frac{25}{16} on both sides

$$-u^2 = -\frac{63}{2}-\frac{25}{16} = -\frac{529}{16}$$

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.

$$r =-\frac{5}{4} - \frac{23}{4} = -7 s = -\frac{5}{4} + \frac{23}{4} = 4.500$$

Evaluate

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