Solve -12x^2+15x+8

$$- 12 x ^ { 2 } + 15 x + 8$$

Factor

$$-12\left(x-\left(-\frac{\sqrt{609}}{24}+\frac{5}{8}\right)\right)\left(x-\left(\frac{\sqrt{609}}{24}+\frac{5}{8}\right)\right)$$

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.

$$-12x^{2}+15x+8=0$$

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.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-12\right)\times 8}}{2\left(-12\right)}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\left(-12\right)\times 8}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-15±\sqrt{225+48\times 8}}{2\left(-12\right)}$$

Multiply 48 times 8.

$$x=\frac{-15±\sqrt{225+384}}{2\left(-12\right)}$$

Add 225 to 384.

$$x=\frac{-15±\sqrt{609}}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{-15±\sqrt{609}}{-24}$$

Now solve the equation x=\frac{-15±\sqrt{609}}{-24} when ± is plus. Add -15 to \sqrt{609}.

$$x=\frac{\sqrt{609}-15}{-24}$$

Divide -15+\sqrt{609} by -24.

$$x=-\frac{\sqrt{609}}{24}+\frac{5}{8}$$

Now solve the equation x=\frac{-15±\sqrt{609}}{-24} when ± is minus. Subtract \sqrt{609} from -15.

$$x=\frac{-\sqrt{609}-15}{-24}$$

Divide -15-\sqrt{609} by -24.

$$x=\frac{\sqrt{609}}{24}+\frac{5}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{8}-\frac{\sqrt{609}}{24} for x_{1} and \frac{5}{8}+\frac{\sqrt{609}}{24} for x_{2}.

$$-12x^{2}+15x+8=-12\left(x-\left(-\frac{\sqrt{609}}{24}+\frac{5}{8}\right)\right)\left(x-\left(\frac{\sqrt{609}}{24}+\frac{5}{8}\right)\right)$$

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.

$$x ^ 2 -\frac{5}{4}x -\frac{2}{3} = 0$$

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

$$r + s = \frac{5}{4} rs = -\frac{2}{3}$$

Two numbers r and s sum up to \frac{5}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{5}{4} = \frac{5}{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>

$$r = \frac{5}{8} - u s = \frac{5}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{2}{3}

$$(\frac{5}{8} - u) (\frac{5}{8} + u) = -\frac{2}{3}$$

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

$$\frac{25}{64} - u^2 = -\frac{2}{3}$$

Simplify the expression by subtracting \frac{25}{64} on both sides

$$-u^2 = -\frac{2}{3}-\frac{25}{64} = -\frac{203}{192}$$

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

$$u^2 = \frac{203}{192} u = \pm\sqrt{\frac{203}{192}} = \pm \frac{\sqrt{203}}{\sqrt{192}}$$

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

$$r =\frac{5}{8} - \frac{\sqrt{203}}{\sqrt{192}} = -0.403 s = \frac{5}{8} + \frac{\sqrt{203}}{\sqrt{192}} = 1.653$$

Evaluate

$$8+15x-12x^{2}$$