Solve 3d^2-3d-2

$$3 d ^ { 2 } - 3 d - 2$$

Factor

$$3\left(d-\left(-\frac{\sqrt{33}}{6}+\frac{1}{2}\right)\right)\left(d-\left(\frac{\sqrt{33}}{6}+\frac{1}{2}\right)\right)$$

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.

$$3d^{2}-3d-2=0$$

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.

$$d=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 3\left(-2\right)}}{2\times 3}$$

Square -3.

$$d=\frac{-\left(-3\right)±\sqrt{9-4\times 3\left(-2\right)}}{2\times 3}$$

Multiply -4 times 3.

$$d=\frac{-\left(-3\right)±\sqrt{9-12\left(-2\right)}}{2\times 3}$$

Multiply -12 times -2.

$$d=\frac{-\left(-3\right)±\sqrt{9+24}}{2\times 3}$$

Add 9 to 24.

$$d=\frac{-\left(-3\right)±\sqrt{33}}{2\times 3}$$

The opposite of -3 is 3.

$$d=\frac{3±\sqrt{33}}{2\times 3}$$

Multiply 2 times 3.

$$d=\frac{3±\sqrt{33}}{6}$$

Now solve the equation d=\frac{3±\sqrt{33}}{6} when ± is plus. Add 3 to \sqrt{33}.

$$d=\frac{\sqrt{33}+3}{6}$$

Divide 3+\sqrt{33} by 6.

$$d=\frac{\sqrt{33}}{6}+\frac{1}{2}$$

Now solve the equation d=\frac{3±\sqrt{33}}{6} when ± is minus. Subtract \sqrt{33} from 3.

$$d=\frac{3-\sqrt{33}}{6}$$

Divide 3-\sqrt{33} by 6.

$$d=-\frac{\sqrt{33}}{6}+\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2}+\frac{\sqrt{33}}{6} for x_{1} and \frac{1}{2}-\frac{\sqrt{33}}{6} for x_{2}.

$$3d^{2}-3d-2=3\left(d-\left(\frac{\sqrt{33}}{6}+\frac{1}{2}\right)\right)\left(d-\left(-\frac{\sqrt{33}}{6}+\frac{1}{2}\right)\right)$$

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.This is achieved by dividing both sides of the equation by 3

$$x ^ 2 -1x -\frac{2}{3} = 0$$

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

$$r + s = 1 rs = -\frac{2}{3}$$

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

$$r = \frac{1}{2} - u s = \frac{1}{2} + u$$

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

$$(\frac{1}{2} - u) (\frac{1}{2} + u) = -\frac{2}{3}$$

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

$$\frac{1}{4} - u^2 = -\frac{2}{3}$$

Simplify the expression by subtracting \frac{1}{4} on both sides

$$-u^2 = -\frac{2}{3}-\frac{1}{4} = -\frac{11}{12}$$

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

$$u^2 = \frac{11}{12} u = \pm\sqrt{\frac{11}{12}} = \pm \frac{\sqrt{11}}{\sqrt{12}}$$

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

$$r =\frac{1}{2} - \frac{\sqrt{11}}{\sqrt{12}} = -0.457 s = \frac{1}{2} + \frac{\sqrt{11}}{\sqrt{12}} = 1.457$$

Evaluate

$$3d^{2}-3d-2$$