Solve 2v^2-18v+28

$$2 v ^ { 2 } - 18 v + 28$$

Factor

$$2\left(v-7\right)\left(v-2\right)$$

Solution Steps

Factor out 2.

$$2\left(v^{2}-9v+14\right)$$

Consider v^{2}-9v+14. Factor the expression by grouping. First, the expression needs to be rewritten as v^{2}+av+bv+14. To find a and b, set up a system to be solved.

$$a+b=-9 ab=1\times 14=14$$

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 14.

$$-1,-14 -2,-7$$

Calculate the sum for each pair.

$$-1-14=-15 -2-7=-9$$

The solution is the pair that gives sum -9.

$$a=-7 b=-2$$

Rewrite v^{2}-9v+14 as \left(v^{2}-7v\right)+\left(-2v+14\right).

$$\left(v^{2}-7v\right)+\left(-2v+14\right)$$

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

$$v\left(v-7\right)-2\left(v-7\right)$$

Factor out common term v-7 by using distributive property.

$$\left(v-7\right)\left(v-2\right)$$

Rewrite the complete factored expression.

$$2\left(v-7\right)\left(v-2\right)$$

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.

$$2v^{2}-18v+28=0$$

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.

$$v=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 2\times 28}}{2\times 2}$$

Square -18.

$$v=\frac{-\left(-18\right)±\sqrt{324-4\times 2\times 28}}{2\times 2}$$

Multiply -4 times 2.

$$v=\frac{-\left(-18\right)±\sqrt{324-8\times 28}}{2\times 2}$$

Multiply -8 times 28.

$$v=\frac{-\left(-18\right)±\sqrt{324-224}}{2\times 2}$$

Add 324 to -224.

$$v=\frac{-\left(-18\right)±\sqrt{100}}{2\times 2}$$

Take the square root of 100.

$$v=\frac{-\left(-18\right)±10}{2\times 2}$$

The opposite of -18 is 18.

$$v=\frac{18±10}{2\times 2}$$

Multiply 2 times 2.

$$v=\frac{18±10}{4}$$

Now solve the equation v=\frac{18±10}{4} when ± is plus. Add 18 to 10.

$$v=\frac{28}{4}$$

Divide 28 by 4.

$$v=7$$

Now solve the equation v=\frac{18±10}{4} when ± is minus. Subtract 10 from 18.

$$v=\frac{8}{4}$$

Divide 8 by 4.

$$v=2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 7 for x_{1} and 2 for x_{2}.

$$2v^{2}-18v+28=2\left(v-7\right)\left(v-2\right)$$

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 2

$$x ^ 2 -9x +14 = 0$$

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

$$r + s = 9 rs = 14$$

Two numbers r and s sum up to 9 exactly when the average of the two numbers is \frac{1}{2}*9 = \frac{9}{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>

$$r = \frac{9}{2} - u s = \frac{9}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 14

$$(\frac{9}{2} - u) (\frac{9}{2} + u) = 14$$

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

$$\frac{81}{4} - u^2 = 14$$

Simplify the expression by subtracting \frac{81}{4} on both sides

$$-u^2 = 14-\frac{81}{4} = -\frac{25}{4}$$

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

$$u^2 = \frac{25}{4} u = \pm\sqrt{\frac{25}{4}} = \pm \frac{5}{2}$$

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

$$r =\frac{9}{2} - \frac{5}{2} = 2 s = \frac{9}{2} + \frac{5}{2} = 7$$

Evaluate

$$2\left(v-7\right)\left(v-2\right)$$