Solve F(x)=6x^2-30x-9

$$F ( x ) = 6 x ^ { 2 } - 30 x - 9$$

Factor

$$6\left(x-\frac{5-\sqrt{31}}{2}\right)\left(x-\frac{\sqrt{31}+5}{2}\right)$$

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.

$$6x^{2}-30x-9=0$$

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.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 6\left(-9\right)}}{2\times 6}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 6\left(-9\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-30\right)±\sqrt{900-24\left(-9\right)}}{2\times 6}$$

Multiply -24 times -9.

$$x=\frac{-\left(-30\right)±\sqrt{900+216}}{2\times 6}$$

Add 900 to 216.

$$x=\frac{-\left(-30\right)±\sqrt{1116}}{2\times 6}$$

Take the square root of 1116.

$$x=\frac{-\left(-30\right)±6\sqrt{31}}{2\times 6}$$

The opposite of -30 is 30.

$$x=\frac{30±6\sqrt{31}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{30±6\sqrt{31}}{12}$$

Now solve the equation x=\frac{30±6\sqrt{31}}{12} when ± is plus. Add 30 to 6\sqrt{31}.

$$x=\frac{6\sqrt{31}+30}{12}$$

Divide 30+6\sqrt{31} by 12.

$$x=\frac{\sqrt{31}+5}{2}$$

Now solve the equation x=\frac{30±6\sqrt{31}}{12} when ± is minus. Subtract 6\sqrt{31} from 30.

$$x=\frac{30-6\sqrt{31}}{12}$$

Divide 30-6\sqrt{31} by 12.

$$x=\frac{5-\sqrt{31}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5+\sqrt{31}}{2} for x_{1} and \frac{5-\sqrt{31}}{2} for x_{2}.

$$6x^{2}-30x-9=6\left(x-\frac{\sqrt{31}+5}{2}\right)\left(x-\frac{5-\sqrt{31}}{2}\right)$$

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 6

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

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

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

Two numbers r and s sum up to 5 exactly when the average of the two numbers is \frac{1}{2}*5 = \frac{5}{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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

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

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

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

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

$$\frac{25}{4} - u^2 = -\frac{3}{2}$$

Simplify the expression by subtracting \frac{25}{4} on both sides

$$-u^2 = -\frac{3}{2}-\frac{25}{4} = -\frac{31}{4}$$

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

$$u^2 = \frac{31}{4} u = \pm\sqrt{\frac{31}{4}} = \pm \frac{\sqrt{31}}{2}$$

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

$$r =\frac{5}{2} - \frac{\sqrt{31}}{2} = -0.284 s = \frac{5}{2} + \frac{\sqrt{31}}{2} = 5.284$$

Evaluate

$$6x^{2}-30x-9$$