Solve f(x)=-2x^2-12x-9

$$f ( x ) = - 2 x ^ { 2 } - 12 x - 9$$

Factor

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

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.

$$-2x^{2}-12x-9=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-2\right)\left(-9\right)}}{2\left(-2\right)}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-2\right)\left(-9\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-12\right)±\sqrt{144+8\left(-9\right)}}{2\left(-2\right)}$$

Multiply 8 times -9.

$$x=\frac{-\left(-12\right)±\sqrt{144-72}}{2\left(-2\right)}$$

Add 144 to -72.

$$x=\frac{-\left(-12\right)±\sqrt{72}}{2\left(-2\right)}$$

Take the square root of 72.

$$x=\frac{-\left(-12\right)±6\sqrt{2}}{2\left(-2\right)}$$

The opposite of -12 is 12.

$$x=\frac{12±6\sqrt{2}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{12±6\sqrt{2}}{-4}$$

Now solve the equation x=\frac{12±6\sqrt{2}}{-4} when ± is plus. Add 12 to 6\sqrt{2}.

$$x=\frac{6\sqrt{2}+12}{-4}$$

Divide 12+6\sqrt{2} by -4.

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

Now solve the equation x=\frac{12±6\sqrt{2}}{-4} when ± is minus. Subtract 6\sqrt{2} from 12.

$$x=\frac{12-6\sqrt{2}}{-4}$$

Divide 12-6\sqrt{2} by -4.

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

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

$$-2x^{2}-12x-9=-2\left(x-\left(-\frac{3\sqrt{2}}{2}-3\right)\right)\left(x-\left(\frac{3\sqrt{2}}{2}-3\right)\right)$$

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.

$$x ^ 2 +6x +\frac{9}{2} = 0$$

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

$$r + s = -6 rs = \frac{9}{2}$$

Two numbers r and s sum up to -6 exactly when the average of the two numbers is \frac{1}{2}*-6 = -3. 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>

$$r = -3 - u s = -3 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{9}{2}

$$(-3 - u) (-3 + u) = \frac{9}{2}$$

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

$$9 - u^2 = \frac{9}{2}$$

Simplify the expression by subtracting 9 on both sides

$$-u^2 = \frac{9}{2}-9 = -\frac{9}{2}$$

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

$$u^2 = \frac{9}{2} u = \pm\sqrt{\frac{9}{2}} = \pm \frac{3}{\sqrt{2}}$$

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

$$r =-3 - \frac{3}{\sqrt{2}} = -5.121 s = -3 + \frac{3}{\sqrt{2}} = -0.879$$

Evaluate

$$-2x^{2}-12x-9$$