Questions: Solve by using the quadratic formula. 6x^2 - 8x = 64 4,-8/3 4,-9/2 14/3,-3 -5 ± i sqrt(479)/6

Transcript text: Solve by using the quadratic formula. \[ 6 x^{2}-8 x=64 \] $\left\{4,-\frac{8}{3}\right\}$ $\left\{4,-\frac{9}{2}\right\}$ $\left\{\frac{14}{3},-3\right\}$ $\left\{\frac{-5 \pm i \sqrt{479}}{6}\right\}$

Solution

Solution Steps

To solve the quadratic equation $6x^2 - 8x = 64$ using the quadratic formula, we first need to rewrite the equation in the standard form $ax^2 + bx + c = 0$. Then, we identify the coefficients $a$, $b$, and $c$ and apply the quadratic formula:

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

Solution Approach

Rewrite the equation in the form $ax^2 + bx + c = 0$.
Identify the coefficients $a$, $b$, and $c$.
Use the quadratic formula to find the solutions for $x$.

Step 1: Rewrite the Equation

We start with the equation: \[ 6x^2 - 8x = 64 \] Rearranging it into standard form gives: \[ 6x^2 - 8x - 64 = 0 \]

Step 2: Identify Coefficients

From the standard form $ax^2 + bx + c = 0$, we identify the coefficients: \[ a = 6, \quad b = -8, \quad c = -64 \]

Step 3: Calculate the Discriminant

We calculate the discriminant using the formula $D = b^2 - 4ac$: \[ D = (-8)^2 - 4 \cdot 6 \cdot (-64) = 64 + 1536 = 1600 \]

Step 4: Apply the Quadratic Formula

Using the quadratic formula $x = \frac{-b \pm \sqrt{D}}{2a}$, we find the solutions: \[ x = \frac{-(-8) \pm \sqrt{1600}}{2 \cdot 6} = \frac{8 \pm 40}{12} \] Calculating the two possible values:

For the positive root: \[ x_1 = \frac{8 + 40}{12} = \frac{48}{12} = 4 \]
For the negative root: \[ x_2 = \frac{8 - 40}{12} = \frac{-32}{12} = -\frac{8}{3} \]