Solve w^2-5w-84

$$w ^ { 2 } - 5 w - 84$$

Factor

$$\left(w-12\right)\left(w+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as w^{2}+aw+bw-84. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\left(-84\right)=-84$$

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 -84.

$$1,-84 2,-42 3,-28 4,-21 6,-14 7,-12$$

Calculate the sum for each pair.

$$1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5$$

The solution is the pair that gives sum -5.

$$a=-12 b=7$$

Rewrite w^{2}-5w-84 as \left(w^{2}-12w\right)+\left(7w-84\right).

$$\left(w^{2}-12w\right)+\left(7w-84\right)$$

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

$$w\left(w-12\right)+7\left(w-12\right)$$

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

$$\left(w-12\right)\left(w+7\right)$$

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.

$$w^{2}-5w-84=0$$

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.

$$w=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-84\right)}}{2}$$

Square -5.

$$w=\frac{-\left(-5\right)±\sqrt{25-4\left(-84\right)}}{2}$$

Multiply -4 times -84.

$$w=\frac{-\left(-5\right)±\sqrt{25+336}}{2}$$

Add 25 to 336.

$$w=\frac{-\left(-5\right)±\sqrt{361}}{2}$$

Take the square root of 361.

$$w=\frac{-\left(-5\right)±19}{2}$$

The opposite of -5 is 5.

$$w=\frac{5±19}{2}$$

Now solve the equation w=\frac{5±19}{2} when ± is plus. Add 5 to 19.

$$w=\frac{24}{2}$$

Divide 24 by 2.

$$w=12$$

Now solve the equation w=\frac{5±19}{2} when ± is minus. Subtract 19 from 5.

$$w=-\frac{14}{2}$$

Divide -14 by 2.

$$w=-7$$

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 -7 for x_{2}.

$$w^{2}-5w-84=\left(w-12\right)\left(w-\left(-7\right)\right)$$

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

$$w^{2}-5w-84=\left(w-12\right)\left(w+7\right)$$

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.

$$x ^ 2 -5x -84 = 0$$

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

$$r + s = 5 rs = -84$$

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

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

To solve for unknown quantity u, substitute these in the product equation rs = -84

$$(\frac{5}{2} - u) (\frac{5}{2} + u) = -84$$

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

$$\frac{25}{4} - u^2 = -84$$

Simplify the expression by subtracting \frac{25}{4} on both sides

$$-u^2 = -84-\frac{25}{4} = -\frac{361}{4}$$

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

$$u^2 = \frac{361}{4} u = \pm\sqrt{\frac{361}{4}} = \pm \frac{19}{2}$$

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

$$r =\frac{5}{2} - \frac{19}{2} = -7 s = \frac{5}{2} + \frac{19}{2} = 12$$

Evaluate

$$\left(w-12\right)\left(w+7\right)$$