Solve 3v^2-51v+90

$$3 v ^ { 2 } - 51 v + 90$$

Factor

$$3\left(v-15\right)\left(v-2\right)$$

Solution Steps

Factor out 3.

$$3\left(v^{2}-17v+30\right)$$

Consider v^{2}-17v+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.

$$a+b=-17 ab=1\times 30=30$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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 -17.

$$a=-15 b=-2$$

Rewrite v^{2}-17v+30 as \left(v^{2}-15v\right)+\left(-2v+30\right).

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

Factor out v in the first and -2 in the second group.

$$v\left(v-15\right)-2\left(v-15\right)$$

Factor out common term v-15 by using distributive property.

$$\left(v-15\right)\left(v-2\right)$$

Rewrite the complete factored expression.

$$3\left(v-15\right)\left(v-2\right)$$

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.

$$3v^{2}-51v+90=0$$

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.

$$v=\frac{-\left(-51\right)±\sqrt{\left(-51\right)^{2}-4\times 3\times 90}}{2\times 3}$$

Square -51.

$$v=\frac{-\left(-51\right)±\sqrt{2601-4\times 3\times 90}}{2\times 3}$$

Multiply -4 times 3.

$$v=\frac{-\left(-51\right)±\sqrt{2601-12\times 90}}{2\times 3}$$

Multiply -12 times 90.

$$v=\frac{-\left(-51\right)±\sqrt{2601-1080}}{2\times 3}$$

Add 2601 to -1080.

$$v=\frac{-\left(-51\right)±\sqrt{1521}}{2\times 3}$$

Take the square root of 1521.

$$v=\frac{-\left(-51\right)±39}{2\times 3}$$

The opposite of -51 is 51.

$$v=\frac{51±39}{2\times 3}$$

Multiply 2 times 3.

$$v=\frac{51±39}{6}$$

Now solve the equation v=\frac{51±39}{6} when ± is plus. Add 51 to 39.

$$v=\frac{90}{6}$$

Divide 90 by 6.

$$v=15$$

Now solve the equation v=\frac{51±39}{6} when ± is minus. Subtract 39 from 51.

$$v=\frac{12}{6}$$

Divide 12 by 6.

$$v=2$$

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}.

$$3v^{2}-51v+90=3\left(v-15\right)\left(v-2\right)$$

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.This is achieved by dividing both sides of the equation by 3

$$x ^ 2 -17x +30 = 0$$

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

$$r + s = 17 rs = 30$$

Two numbers r and s sum up to 17 exactly when the average of the two numbers is \frac{1}{2}*17 = \frac{17}{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>

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

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

$$(\frac{17}{2} - u) (\frac{17}{2} + u) = 30$$

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

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

Simplify the expression by subtracting \frac{289}{4} on both sides

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

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

$$u^2 = \frac{169}{4} u = \pm\sqrt{\frac{169}{4}} = \pm \frac{13}{2}$$

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

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

Evaluate

$$3\left(v-15\right)\left(v-2\right)$$