Questions: g(x)=-4x^2-12x+9

Solution Steps

To find the roots of the quadratic function $g(x) = -4x^2 - 12x + 9$, we can use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = -4$, $b = -12$, and $c = 9$.

We are given the quadratic function $g(x) = -4x^2 - 12x + 9$. To find the roots of this function, we will use the quadratic formula.

The discriminant $D$ is calculated using the formula: $$D = b^2 - 4ac$$ Substituting the values $a = -4$, $b = -12$, and $c = 9$: $$D = (-12)^2 - 4(-4)(9) = 144 + 144 = 288$$

Using the quadratic formula: $$x = \frac{-b \pm \sqrt{D}}{2a}$$ we substitute $b = -12$, $D = 288$, and $a = -4$: $$x = \frac{12 \pm \sqrt{288}}{-8}$$ Calculating the square root: $$\sqrt{288} = 16.9706 \quad (\text{rounded to four significant digits})$$ Thus, the roots are: $$x_1 = \frac{12 + 16.9706}{-8} \approx -3.6213$$ $$x_2 = \frac{12 - 16.9706}{-8} \approx 0.6213$$

Final Answer