Solve 8v^2-15v-18

$$8 v ^ { 2 } - 15 v - 18$$

Factor

$$8\left(v-\frac{15-3\sqrt{89}}{16}\right)\left(v-\frac{3\sqrt{89}+15}{16}\right)$$

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.

$$8v^{2}-15v-18=0$$

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.

$$v=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 8\left(-18\right)}}{2\times 8}$$

Square -15.

$$v=\frac{-\left(-15\right)±\sqrt{225-4\times 8\left(-18\right)}}{2\times 8}$$

Multiply -4 times 8.

$$v=\frac{-\left(-15\right)±\sqrt{225-32\left(-18\right)}}{2\times 8}$$

Multiply -32 times -18.

$$v=\frac{-\left(-15\right)±\sqrt{225+576}}{2\times 8}$$

Add 225 to 576.

$$v=\frac{-\left(-15\right)±\sqrt{801}}{2\times 8}$$

Take the square root of 801.

$$v=\frac{-\left(-15\right)±3\sqrt{89}}{2\times 8}$$

The opposite of -15 is 15.

$$v=\frac{15±3\sqrt{89}}{2\times 8}$$

Multiply 2 times 8.

$$v=\frac{15±3\sqrt{89}}{16}$$

Now solve the equation v=\frac{15±3\sqrt{89}}{16} when ± is plus. Add 15 to 3\sqrt{89}.

$$v=\frac{3\sqrt{89}+15}{16}$$

Now solve the equation v=\frac{15±3\sqrt{89}}{16} when ± is minus. Subtract 3\sqrt{89} from 15.

$$v=\frac{15-3\sqrt{89}}{16}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{15+3\sqrt{89}}{16} for x_{1} and \frac{15-3\sqrt{89}}{16} for x_{2}.

$$8v^{2}-15v-18=8\left(v-\frac{3\sqrt{89}+15}{16}\right)\left(v-\frac{15-3\sqrt{89}}{16}\right)$$

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

$$x ^ 2 -\frac{15}{8}x -\frac{9}{4} = 0$$

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

$$r + s = \frac{15}{8} rs = -\frac{9}{4}$$

Two numbers r and s sum up to \frac{15}{8} exactly when the average of the two numbers is \frac{1}{2}*\frac{15}{8} = \frac{15}{16}. 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>

$$r = \frac{15}{16} - u s = \frac{15}{16} + u$$

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

$$(\frac{15}{16} - u) (\frac{15}{16} + u) = -\frac{9}{4}$$

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

$$\frac{225}{256} - u^2 = -\frac{9}{4}$$

Simplify the expression by subtracting \frac{225}{256} on both sides

$$-u^2 = -\frac{9}{4}-\frac{225}{256} = -\frac{801}{256}$$

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

$$u^2 = \frac{801}{256} u = \pm\sqrt{\frac{801}{256}} = \pm \frac{\sqrt{801}}{16}$$

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

$$r =\frac{15}{16} - \frac{\sqrt{801}}{16} = -0.831 s = \frac{15}{16} + \frac{\sqrt{801}}{16} = 2.706$$

Evaluate

$$8v^{2}-15v-18$$