Solve 3v^2+33v+90

$$3 v ^ { 2 } + 33 v + 90$$

Factor

$$3\left(v+5\right)\left(v+6\right)$$

Solution Steps

Factor out 3.

$$3\left(v^{2}+11v+30\right)$$

Consider v^{2}+11v+30. Factor the expression by grouping. First, the expression needs to be rewritten as v^{2}+av+bv+30. To find a and b, set up a system to be solved.

$$a+b=11 ab=1\times 30=30$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 30.

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

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 11.

$$a=5 b=6$$

Rewrite v^{2}+11v+30 as \left(v^{2}+5v\right)+\left(6v+30\right).

$$\left(v^{2}+5v\right)+\left(6v+30\right)$$

Factor out v in the first and 6 in the second group.

$$v\left(v+5\right)+6\left(v+5\right)$$

Factor out common term v+5 by using distributive property.

$$\left(v+5\right)\left(v+6\right)$$

Rewrite the complete factored expression.

$$3\left(v+5\right)\left(v+6\right)$$

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.

$$3v^{2}+33v+90=0$$

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.

$$v=\frac{-33±\sqrt{33^{2}-4\times 3\times 90}}{2\times 3}$$

Square 33.

$$v=\frac{-33±\sqrt{1089-4\times 3\times 90}}{2\times 3}$$

Multiply -4 times 3.

$$v=\frac{-33±\sqrt{1089-12\times 90}}{2\times 3}$$

Multiply -12 times 90.

$$v=\frac{-33±\sqrt{1089-1080}}{2\times 3}$$

Add 1089 to -1080.

$$v=\frac{-33±\sqrt{9}}{2\times 3}$$

Take the square root of 9.

$$v=\frac{-33±3}{2\times 3}$$

Multiply 2 times 3.

$$v=\frac{-33±3}{6}$$

Now solve the equation v=\frac{-33±3}{6} when ± is plus. Add -33 to 3.

$$v=-\frac{30}{6}$$

Divide -30 by 6.

$$v=-5$$

Now solve the equation v=\frac{-33±3}{6} when ± is minus. Subtract 3 from -33.

$$v=-\frac{36}{6}$$

Divide -36 by 6.

$$v=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -5 for x_{1} and -6 for x_{2}.

$$3v^{2}+33v+90=3\left(v-\left(-5\right)\right)\left(v-\left(-6\right)\right)$$

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

$$3v^{2}+33v+90=3\left(v+5\right)\left(v+6\right)$$

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 3

$$x ^ 2 +11x +30 = 0$$

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

$$r + s = -11 rs = 30$$

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

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

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

$$(-\frac{11}{2} - u) (-\frac{11}{2} + u) = 30$$

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

$$\frac{121}{4} - u^2 = 30$$

Simplify the expression by subtracting \frac{121}{4} on both sides

$$-u^2 = 30-\frac{121}{4} = -\frac{1}{4}$$

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

$$u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{2}$$

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

$$r =-\frac{11}{2} - \frac{1}{2} = -6 s = -\frac{11}{2} + \frac{1}{2} = -5$$

Evaluate

$$3\left(v+5\right)\left(v+6\right)$$