Final answer:
To solve the equation 2k^2 + 3k + 15 = 7 using the quadratic formula, rearrange the equation to have 0 on one side, and then apply the quadratic formula using the coefficients a = 2, b = 3, and c = 8. Simplify and evaluate the square root to find the solutions. In this case, the equation has no real solutions.
Step-by-step explanation:
To solve the equation 2k^2 + 3k + 15 = 7 using the quadratic formula, we need to rearrange the equation to have 0 on one side:
2k^2 + 3k + 15 - 7 = 0
2k^2 + 3k + 8 = 0
Now, we can use the quadratic formula, which is given by:
k = (-b ± sqrt(b^2 - 4ac)) / (2a)
Identifying the coefficients, a = 2, b = 3, and c = 8, we can substitute these values into the formula:
k = (-3 ± sqrt(3^2 - 4(2)(8))) / (2(2))
Simplifying gives us two solutions:
k = (-3 + sqrt(9 - 64)) / 4
k = (-3 - sqrt(9 - 64)) / 4
After evaluating the square root, we get:
k = (-3 + sqrt(-55)) / 4 (no real solutions)
k = (-3 - sqrt(-55)) / 4 (no real solutions)
Therefore, the equation 2k^2 + 3k + 15 = 7 has no real solutions.