Solve 14v^2-23v+8=0

$$14 v ^ { 2 } - 23 v + 8 = 0$$

Solve for v

$$v=\frac{1}{2}=0.5$$

$$v = \frac{8}{7} = 1\frac{1}{7} \approx 1.142857143$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 14v^{2}+av+bv+8. To find a and b, set up a system to be solved.

$$a+b=-23 ab=14\times 8=112$$

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 112.

$$-1,-112 -2,-56 -4,-28 -7,-16 -8,-14$$

Calculate the sum for each pair.

$$-1-112=-113 -2-56=-58 -4-28=-32 -7-16=-23 -8-14=-22$$

The solution is the pair that gives sum -23.

$$a=-16 b=-7$$

Rewrite 14v^{2}-23v+8 as \left(14v^{2}-16v\right)+\left(-7v+8\right).

$$\left(14v^{2}-16v\right)+\left(-7v+8\right)$$

Factor out 2v in the first and -1 in the second group.

$$2v\left(7v-8\right)-\left(7v-8\right)$$

Factor out common term 7v-8 by using distributive property.

$$\left(7v-8\right)\left(2v-1\right)$$

To find equation solutions, solve 7v-8=0 and 2v-1=0.

$$v=\frac{8}{7} v=\frac{1}{2}$$

Steps Using the Quadratic Formula

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.

$$14v^{2}-23v+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 14 for a, -23 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 14\times 8}}{2\times 14}$$

Square -23.

$$v=\frac{-\left(-23\right)±\sqrt{529-4\times 14\times 8}}{2\times 14}$$

Multiply -4 times 14.

$$v=\frac{-\left(-23\right)±\sqrt{529-56\times 8}}{2\times 14}$$

Multiply -56 times 8.

$$v=\frac{-\left(-23\right)±\sqrt{529-448}}{2\times 14}$$

Add 529 to -448.

$$v=\frac{-\left(-23\right)±\sqrt{81}}{2\times 14}$$

Take the square root of 81.

$$v=\frac{-\left(-23\right)±9}{2\times 14}$$

The opposite of -23 is 23.

$$v=\frac{23±9}{2\times 14}$$

Multiply 2 times 14.

$$v=\frac{23±9}{28}$$

Now solve the equation v=\frac{23±9}{28} when ± is plus. Add 23 to 9.

$$v=\frac{32}{28}$$

Reduce the fraction \frac{32}{28} to lowest terms by extracting and canceling out 4.

$$v=\frac{8}{7}$$

Now solve the equation v=\frac{23±9}{28} when ± is minus. Subtract 9 from 23.

$$v=\frac{14}{28}$$

Reduce the fraction \frac{14}{28} to lowest terms by extracting and canceling out 14.

$$v=\frac{1}{2}$$

The equation is now solved.

$$v=\frac{8}{7} v=\frac{1}{2}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$14v^{2}-23v+8=0$$

Subtract 8 from both sides of the equation.

$$14v^{2}-23v+8-8=-8$$

Subtracting 8 from itself leaves 0.

$$14v^{2}-23v=-8$$

Divide both sides by 14.

$$\frac{14v^{2}-23v}{14}=-\frac{8}{14}$$

Dividing by 14 undoes the multiplication by 14.

$$v^{2}-\frac{23}{14}v=-\frac{8}{14}$$

Reduce the fraction \frac{-8}{14} to lowest terms by extracting and canceling out 2.

$$v^{2}-\frac{23}{14}v=-\frac{4}{7}$$

Divide -\frac{23}{14}, the coefficient of the x term, by 2 to get -\frac{23}{28}. Then add the square of -\frac{23}{28} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}-\frac{23}{14}v+\left(-\frac{23}{28}\right)^{2}=-\frac{4}{7}+\left(-\frac{23}{28}\right)^{2}$$

Square -\frac{23}{28} by squaring both the numerator and the denominator of the fraction.

$$v^{2}-\frac{23}{14}v+\frac{529}{784}=-\frac{4}{7}+\frac{529}{784}$$

Add -\frac{4}{7} to \frac{529}{784} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$v^{2}-\frac{23}{14}v+\frac{529}{784}=\frac{81}{784}$$

Factor v^{2}-\frac{23}{14}v+\frac{529}{784}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v-\frac{23}{28}\right)^{2}=\frac{81}{784}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-\frac{23}{28}\right)^{2}}=\sqrt{\frac{81}{784}}$$

Simplify.

$$v-\frac{23}{28}=\frac{9}{28} v-\frac{23}{28}=-\frac{9}{28}$$

Add \frac{23}{28} to both sides of the equation.

$$v=\frac{8}{7} v=\frac{1}{2}$$

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 14

$$x ^ 2 -\frac{23}{14}x +\frac{4}{7} = 0$$

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

$$r + s = \frac{23}{14} rs = \frac{4}{7}$$

Two numbers r and s sum up to \frac{23}{14} exactly when the average of the two numbers is \frac{1}{2}*\frac{23}{14} = \frac{23}{28}. 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>

$$r = \frac{23}{28} - u s = \frac{23}{28} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{4}{7}

$$(\frac{23}{28} - u) (\frac{23}{28} + u) = \frac{4}{7}$$

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

$$\frac{529}{784} - u^2 = \frac{4}{7}$$

Simplify the expression by subtracting \frac{529}{784} on both sides

$$-u^2 = \frac{4}{7}-\frac{529}{784} = -\frac{81}{784}$$

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

$$u^2 = \frac{81}{784} u = \pm\sqrt{\frac{81}{784}} = \pm \frac{9}{28}$$

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

$$r =\frac{23}{28} - \frac{9}{28} = 0.500 s = \frac{23}{28} + \frac{9}{28} = 1.143$$