Questions: Solve for a. Answer exactly. 4 - 3/a^2 = 4/a

Solution Steps

To solve the equation \(4 - \frac{3}{a^2} = \frac{4}{a}\) for \(a\), we can follow these steps:

1. Move all terms to one side of the equation to set it to zero.
2. Multiply through by \(a^2\) to clear the denominators.
3. Simplify and solve the resulting quadratic equation for \(a\).

We start with the equation: \[ 4 - \frac{3}{a^2} = \frac{4}{a} \] To eliminate the fractions, we can multiply through by \(a^2\): \[ a^2 \left(4 - \frac{3}{a^2}\right) = a^2 \left(\frac{4}{a}\right) \] This simplifies to: \[ 4a^2 - 3 = 4a \]

Rearranging the equation gives us: \[ 4a^2 - 4a - 3 = 0 \] This is a standard quadratic equation in the form \(Ax^2 + Bx + C = 0\) where \(A = 4\), \(B = -4\), and \(C = -3\).

Using the quadratic formula \(a = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\), we can find the values of \(a\): \[ a = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 4 \cdot (-3)}}{2 \cdot 4} \] Calculating the discriminant: \[ (-4)^2 - 4 \cdot 4 \cdot (-3) = 16 + 48 = 64 \] Thus, we have: \[ a = \frac{4 \pm \sqrt{64}}{8} = \frac{4 \pm 8}{8} \] This results in two possible solutions: \[ a = \frac{12}{8} = \frac{3}{2} \quad \text{and} \quad a = \frac{-4}{8} = -\frac{1}{2} \]

Final Answer