Questions: Solve by the quadratic formula. List the solutions, separated by commas. 12 x^2+11 x+2=0 x=

Transcript text: Solve by the quadratic formula. List the solutions, separated by commas. \[ 12 x^{2}+11 x+2=0 \] \[ x= \]

Solution

Solution Steps

To solve the quadratic equation \(12x^2 + 11x + 2 = 0\) using the quadratic formula, we will identify the coefficients \(a\), \(b\), and \(c\) from the equation. The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

We will substitute the values of \(a\), \(b\), and \(c\) into this formula to find the solutions for \(x\).

Step 1: Identify Coefficients

For the quadratic equation \(12x^2 + 11x + 2 = 0\), we identify the coefficients as follows:

\(a = 12\)
\(b = 11\)
\(c = 2\)

Step 2: Calculate the Discriminant

We calculate the discriminant \(D\) using the formula: \[ D = b^2 - 4ac \] Substituting the values: \[ D = 11^2 - 4 \cdot 12 \cdot 2 = 121 - 96 = 25 \]

Step 3: Apply the Quadratic Formula

Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] we find the two solutions: \[ x_1 = \frac{-11 + \sqrt{25}}{2 \cdot 12} = \frac{-11 + 5}{24} = \frac{-6}{24} = -0.25 \] \[ x_2 = \frac{-11 - \sqrt{25}}{2 \cdot 12} = \frac{-11 - 5}{24} = \frac{-16}{24} = -\frac{2}{3} \approx -0.6667 \]