Solve f(x)=-3x^2-9x+8

$$f ( x ) = - 3 x ^ { 2 } - 9 x + 8$$

Factor

$$-3\left(x-\left(-\frac{\sqrt{177}}{6}-\frac{3}{2}\right)\right)\left(x-\left(\frac{\sqrt{177}}{6}-\frac{3}{2}\right)\right)$$

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.

$$-3x^{2}-9x+8=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-3\right)\times 8}}{2\left(-3\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-3\right)\times 8}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-9\right)±\sqrt{81+12\times 8}}{2\left(-3\right)}$$

Multiply 12 times 8.

$$x=\frac{-\left(-9\right)±\sqrt{81+96}}{2\left(-3\right)}$$

Add 81 to 96.

$$x=\frac{-\left(-9\right)±\sqrt{177}}{2\left(-3\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{177}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{9±\sqrt{177}}{-6}$$

Now solve the equation x=\frac{9±\sqrt{177}}{-6} when ± is plus. Add 9 to \sqrt{177}.

$$x=\frac{\sqrt{177}+9}{-6}$$

Divide 9+\sqrt{177} by -6.

$$x=-\frac{\sqrt{177}}{6}-\frac{3}{2}$$

Now solve the equation x=\frac{9±\sqrt{177}}{-6} when ± is minus. Subtract \sqrt{177} from 9.

$$x=\frac{9-\sqrt{177}}{-6}$$

Divide 9-\sqrt{177} by -6.

$$x=\frac{\sqrt{177}}{6}-\frac{3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{2}-\frac{\sqrt{177}}{6} for x_{1} and -\frac{3}{2}+\frac{\sqrt{177}}{6} for x_{2}.

$$-3x^{2}-9x+8=-3\left(x-\left(-\frac{\sqrt{177}}{6}-\frac{3}{2}\right)\right)\left(x-\left(\frac{\sqrt{177}}{6}-\frac{3}{2}\right)\right)$$

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.

$$x ^ 2 +3x -\frac{8}{3} = 0$$

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

$$r + s = -3 rs = -\frac{8}{3}$$

Two numbers r and s sum up to -3 exactly when the average of the two numbers is \frac{1}{2}*-3 = -\frac{3}{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>

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

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{8}{3}

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

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

$$\frac{9}{4} - u^2 = -\frac{8}{3}$$

Simplify the expression by subtracting \frac{9}{4} on both sides

$$-u^2 = -\frac{8}{3}-\frac{9}{4} = -\frac{59}{12}$$

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

$$u^2 = \frac{59}{12} u = \pm\sqrt{\frac{59}{12}} = \pm \frac{\sqrt{59}}{\sqrt{12}}$$

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

$$r =-\frac{3}{2} - \frac{\sqrt{59}}{\sqrt{12}} = -3.717 s = -\frac{3}{2} + \frac{\sqrt{59}}{\sqrt{12}} = 0.717$$

Evaluate

$$8-9x-3x^{2}$$