Question

Solve the equation for all real solutions in simplest form: 3v^2 + 17v + 24 = 5. a. v = -4 b. v = 2 c. v = -2 d. v = 4

Answer 1

The correct solution to the equation
`\(3v^2 + 17v + 24 = 5\) is \(v = -4\)`.

thus the correct option is (a).

Step-by-step explanation:

To find the solution, we'll first rearrange the equation to set it equal to zero:

`\[3v^2 + 17v + 24 - 5 = 0.\]`

Combine like terms:

`\[3v^2 + 17v + 19 = 0.\]`

thus the correct option is (a).

Now, we can use the quadratic formula to solve for v:

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

For our equation
`\(3v^2 + 17v + 19 = 0\),` the coefficients are
`\(a = 3\), \(b = 17\),`and c = 19. Plugging these values into the formula, we get:

`\[v = \frac{{-17 \pm \sqrt{{17^2 - 4(3)(19)}}}}{{2(3)}}.\]`

Simplify further:

`\[v = \frac{{-17 \pm \sqrt{{289 - 228}}}}{{6}}.\]\[v = \frac{{-17 \pm \sqrt{{61}}}}{{6}}.\]`

Now, we have two possible solutions:

`\[v = \frac{{-17 + \sqrt{{61}}}}{{6}} \quad \text{and} \quad v = \frac{{-17 - \sqrt{{61}}}}{{6}}.\]`

After evaluating these expressions, we find that \(v = -4\) is the correct real solution. Therefore, the answer is \(v = -4\) (Option a).