Solve 6v^2+36v-96

$$6 v ^ { 2 } + 36 v - 96$$

Factor

$$6\left(v-2\right)\left(v+8\right)$$

Solution Steps

Factor out 6.

$$6\left(v^{2}+6v-16\right)$$

Consider v^{2}+6v-16. Factor the expression by grouping. First, the expression needs to be rewritten as v^{2}+av+bv-16. To find a and b, set up a system to be solved.

$$a+b=6 ab=1\left(-16\right)=-16$$

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 -16.

$$-1,16 -2,8 -4,4$$

Calculate the sum for each pair.

$$-1+16=15 -2+8=6 -4+4=0$$

The solution is the pair that gives sum 6.

$$a=-2 b=8$$

Rewrite v^{2}+6v-16 as \left(v^{2}-2v\right)+\left(8v-16\right).

$$\left(v^{2}-2v\right)+\left(8v-16\right)$$

Factor out v in the first and 8 in the second group.

$$v\left(v-2\right)+8\left(v-2\right)$$

Factor out common term v-2 by using distributive property.

$$\left(v-2\right)\left(v+8\right)$$

Rewrite the complete factored expression.

$$6\left(v-2\right)\left(v+8\right)$$

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.

$$6v^{2}+36v-96=0$$

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.

$$v=\frac{-36±\sqrt{36^{2}-4\times 6\left(-96\right)}}{2\times 6}$$

Square 36.

$$v=\frac{-36±\sqrt{1296-4\times 6\left(-96\right)}}{2\times 6}$$

Multiply -4 times 6.

$$v=\frac{-36±\sqrt{1296-24\left(-96\right)}}{2\times 6}$$

Multiply -24 times -96.

$$v=\frac{-36±\sqrt{1296+2304}}{2\times 6}$$

Add 1296 to 2304.

$$v=\frac{-36±\sqrt{3600}}{2\times 6}$$

Take the square root of 3600.

$$v=\frac{-36±60}{2\times 6}$$

Multiply 2 times 6.

$$v=\frac{-36±60}{12}$$

Now solve the equation v=\frac{-36±60}{12} when ± is plus. Add -36 to 60.

$$v=\frac{24}{12}$$

Divide 24 by 12.

$$v=2$$

Now solve the equation v=\frac{-36±60}{12} when ± is minus. Subtract 60 from -36.

$$v=-\frac{96}{12}$$

Divide -96 by 12.

$$v=-8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2 for x_{1} and -8 for x_{2}.

$$6v^{2}+36v-96=6\left(v-2\right)\left(v-\left(-8\right)\right)$$

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

$$6v^{2}+36v-96=6\left(v-2\right)\left(v+8\right)$$

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 6

$$x ^ 2 +6x -16 = 0$$

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

$$r + s = -6 rs = -16$$

Two numbers r and s sum up to -6 exactly when the average of the two numbers is \frac{1}{2}*-6 = -3. 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>

$$r = -3 - u s = -3 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -16

$$(-3 - u) (-3 + u) = -16$$

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

$$9 - u^2 = -16$$

Simplify the expression by subtracting 9 on both sides

$$-u^2 = -16-9 = -25$$

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

$$u^2 = 25 u = \pm\sqrt{25} = \pm 5$$

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

$$r =-3 - 5 = -8 s = -3 + 5 = 2$$

Evaluate

$$6\left(v-2\right)\left(v+8\right)$$