Questions: Consider the following quadratic equation: 9x^2 + 34x - 11 = -3

Solution Steps

To solve the quadratic equation \(9x^2 + 34x - 11 = -3\), we first need to move all terms to one side of the equation to set it to zero. Then, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots of the equation.

1. Move all terms to one side to set the equation to zero.
2. Identify the coefficients \(a\), \(b\), and \(c\) from the standard form \(ax^2 + bx + c = 0\).
3. Use the quadratic formula to solve for \(x\).

We start with the quadratic equation: \[ 9x^2 + 34x - 11 = -3 \] Rearranging this gives: \[ 9x^2 + 34x + 8 = 0 \]

From the standard form \(ax^2 + bx + c = 0\), we identify the coefficients:

• \(a = 9\)
• \(b = 34\)
• \(c = 8\)

The discriminant \(D\) is calculated as: \[ D = b^2 - 4ac = 34^2 - 4 \cdot 9 \cdot 8 = 1156 - 288 = 868 \]

Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] we find the two solutions: \[ x_1 = \frac{-34 + \sqrt{868}}{2 \cdot 9} \quad \text{and} \quad x_2 = \frac{-34 - \sqrt{868}}{2 \cdot 9} \] Calculating these gives: \[ x_1 \approx 0.2222 \quad \text{and} \quad x_2 = -4.0 \]

Final Answer