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