Solve -2w^2-26w+60

$$- 2 w ^ { 2 } - 26 w + 60$$

Factor

$$2\left(2-w\right)\left(w+15\right)$$

Solution Steps

Factor out 2.

$$2\left(-w^{2}-13w+30\right)$$

Consider -w^{2}-13w+30. Factor the expression by grouping. First, the expression needs to be rewritten as -w^{2}+aw+bw+30. To find a and b, set up a system to be solved.

$$a+b=-13 ab=-30=-30$$

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 -30.

$$1,-30 2,-15 3,-10 5,-6$$

Calculate the sum for each pair.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

The solution is the pair that gives sum -13.

$$a=2 b=-15$$

Rewrite -w^{2}-13w+30 as \left(-w^{2}+2w\right)+\left(-15w+30\right).

$$\left(-w^{2}+2w\right)+\left(-15w+30\right)$$

Factor out w in the first and 15 in the second group.

$$w\left(-w+2\right)+15\left(-w+2\right)$$

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

$$\left(-w+2\right)\left(w+15\right)$$

Rewrite the complete factored expression.

$$2\left(-w+2\right)\left(w+15\right)$$

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.

$$-2w^{2}-26w+60=0$$

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.

$$w=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\left(-2\right)\times 60}}{2\left(-2\right)}$$

Square -26.

$$w=\frac{-\left(-26\right)±\sqrt{676-4\left(-2\right)\times 60}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$w=\frac{-\left(-26\right)±\sqrt{676+8\times 60}}{2\left(-2\right)}$$

Multiply 8 times 60.

$$w=\frac{-\left(-26\right)±\sqrt{676+480}}{2\left(-2\right)}$$

Add 676 to 480.

$$w=\frac{-\left(-26\right)±\sqrt{1156}}{2\left(-2\right)}$$

Take the square root of 1156.

$$w=\frac{-\left(-26\right)±34}{2\left(-2\right)}$$

The opposite of -26 is 26.

$$w=\frac{26±34}{2\left(-2\right)}$$

Multiply 2 times -2.

$$w=\frac{26±34}{-4}$$

Now solve the equation w=\frac{26±34}{-4} when ± is plus. Add 26 to 34.

$$w=\frac{60}{-4}$$

Divide 60 by -4.

$$w=-15$$

Now solve the equation w=\frac{26±34}{-4} when ± is minus. Subtract 34 from 26.

$$w=-\frac{8}{-4}$$

Divide -8 by -4.

$$w=2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -15 for x_{1} and 2 for x_{2}.

$$-2w^{2}-26w+60=-2\left(w-\left(-15\right)\right)\left(w-2\right)$$

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

$$-2w^{2}-26w+60=-2\left(w+15\right)\left(w-2\right)$$

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.

$$x ^ 2 +13x -30 = 0$$

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

$$r + s = -13 rs = -30$$

Two numbers r and s sum up to -13 exactly when the average of the two numbers is \frac{1}{2}*-13 = -\frac{13}{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>

$$r = -\frac{13}{2} - u s = -\frac{13}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -30

$$(-\frac{13}{2} - u) (-\frac{13}{2} + u) = -30$$

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

$$\frac{169}{4} - u^2 = -30$$

Simplify the expression by subtracting \frac{169}{4} on both sides

$$-u^2 = -30-\frac{169}{4} = -\frac{289}{4}$$

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

$$u^2 = \frac{289}{4} u = \pm\sqrt{\frac{289}{4}} = \pm \frac{17}{2}$$

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

$$r =-\frac{13}{2} - \frac{17}{2} = -15 s = -\frac{13}{2} + \frac{17}{2} = 2$$

Evaluate

$$2\left(2-w\right)\left(w+15\right)$$