Questions: Solve the equation for all real solutions in simplest form. 4 a^2 + a - 6 = 0

Solution Steps

To solve the quadratic equation \(4a^2 + a - 6 = 0\), we can use the quadratic formula, which is given by: \[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\).

In this case, \(a = 4\), \(b = 1\), and \(c = -6\). We will substitute these values into the quadratic formula to find the solutions.

For the quadratic equation \(4a^2 + a - 6 = 0\), we identify the coefficients as follows:

• \(A = 4\)
• \(B = 1\)
• \(C = -6\)

The discriminant \(D\) is calculated using the formula: \[ D = B^2 - 4AC \] Substituting the values, we have: \[ D = 1^2 - 4 \cdot 4 \cdot (-6) = 1 + 96 = 97 \]

Using the quadratic formula: \[ a = \frac{-B \pm \sqrt{D}}{2A} \] we substitute \(B = 1\), \(D = 97\), and \(A = 4\): \[ a = \frac{-1 \pm \sqrt{97}}{2 \cdot 4} = \frac{-1 \pm \sqrt{97}}{8} \]

The two solutions are: \[ a_1 = \frac{-1 + \sqrt{97}}{8} \quad \text{and} \quad a_2 = \frac{-1 - \sqrt{97}}{8} \] Calculating these values gives: \[ a_1 \approx 1.1061 \quad \text{and} \quad a_2 \approx -1.3561 \]

Final Answer