Solve 8v^2-30v+18

$$8 v ^ { 2 } - 30 v + 18$$

Factor

$$2\left(v-3\right)\left(4v-3\right)$$

Solution Steps

Factor out 2.

$$2\left(4v^{2}-15v+9\right)$$

Consider 4v^{2}-15v+9. Factor the expression by grouping. First, the expression needs to be rewritten as 4v^{2}+av+bv+9. To find a and b, set up a system to be solved.

$$a+b=-15 ab=4\times 9=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -15.

$$a=-12 b=-3$$

Rewrite 4v^{2}-15v+9 as \left(4v^{2}-12v\right)+\left(-3v+9\right).

$$\left(4v^{2}-12v\right)+\left(-3v+9\right)$$

Factor out 4v in the first and -3 in the second group.

$$4v\left(v-3\right)-3\left(v-3\right)$$

Factor out common term v-3 by using distributive property.

$$\left(v-3\right)\left(4v-3\right)$$

Rewrite the complete factored expression.

$$2\left(v-3\right)\left(4v-3\right)$$

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.

$$8v^{2}-30v+18=0$$

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.

$$v=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 8\times 18}}{2\times 8}$$

Square -30.

$$v=\frac{-\left(-30\right)±\sqrt{900-4\times 8\times 18}}{2\times 8}$$

Multiply -4 times 8.

$$v=\frac{-\left(-30\right)±\sqrt{900-32\times 18}}{2\times 8}$$

Multiply -32 times 18.

$$v=\frac{-\left(-30\right)±\sqrt{900-576}}{2\times 8}$$

Add 900 to -576.

$$v=\frac{-\left(-30\right)±\sqrt{324}}{2\times 8}$$

Take the square root of 324.

$$v=\frac{-\left(-30\right)±18}{2\times 8}$$

The opposite of -30 is 30.

$$v=\frac{30±18}{2\times 8}$$

Multiply 2 times 8.

$$v=\frac{30±18}{16}$$

Now solve the equation v=\frac{30±18}{16} when ± is plus. Add 30 to 18.

$$v=\frac{48}{16}$$

Divide 48 by 16.

$$v=3$$

Now solve the equation v=\frac{30±18}{16} when ± is minus. Subtract 18 from 30.

$$v=\frac{12}{16}$$

Reduce the fraction \frac{12}{16} to lowest terms by extracting and canceling out 4.

$$v=\frac{3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3 for x_{1} and \frac{3}{4} for x_{2}.

$$8v^{2}-30v+18=8\left(v-3\right)\left(v-\frac{3}{4}\right)$$

Subtract \frac{3}{4} from v by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8v^{2}-30v+18=8\left(v-3\right)\times \frac{4v-3}{4}$$

Cancel out 4, the greatest common factor in 8 and 4.

$$8v^{2}-30v+18=2\left(v-3\right)\left(4v-3\right)$$

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 8

$$x ^ 2 -\frac{15}{4}x +\frac{9}{4} = 0$$

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

$$r + s = \frac{15}{4} rs = \frac{9}{4}$$

Two numbers r and s sum up to \frac{15}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{15}{4} = \frac{15}{8}. 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>

$$r = \frac{15}{8} - u s = \frac{15}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{9}{4}

$$(\frac{15}{8} - u) (\frac{15}{8} + u) = \frac{9}{4}$$

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

$$\frac{225}{64} - u^2 = \frac{9}{4}$$

Simplify the expression by subtracting \frac{225}{64} on both sides

$$-u^2 = \frac{9}{4}-\frac{225}{64} = -\frac{81}{64}$$

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

$$u^2 = \frac{81}{64} u = \pm\sqrt{\frac{81}{64}} = \pm \frac{9}{8}$$

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

$$r =\frac{15}{8} - \frac{9}{8} = 0.750 s = \frac{15}{8} + \frac{9}{8} = 3$$

Evaluate

$$2\left(v-3\right)\left(4v-3\right)$$