Final answer:
To find the value of a in a quadratic equation when one solution is given, we use the fact that complex roots occur in conjugate pairs. Given x = 5 + 3/2i as one solution, the other root is x = 5 - 3/2i. We can compare the quadratic equation with the solved equations to determine that a = -10.
Step-by-step explanation:
The quadratic equation is given by:
x² + 1.2x - 6.0 × 10-3 = 0
To find the value of a if one solution is x = 5 + 3/2i, we can use the fact that complex roots occur in conjugate pairs. Since one root is given as x = 5 + 3/2i, the other root will be its conjugate, which is x = 5 - 3/2i.
Therefore, we can set up two equations:
x = 5 + 3/2i
x = 5 - 3/2i
Now we can solve each equation separately:
For x = 5 + 3/2i:
(x - 5 - 3/2i)(x - 5 + 3/2i) = 0
x² - 5x + (3/2i)x - 5x + 25 - 15/2i - (3/2i)x + 15/2i + 3/2i² = 0
x² - 10x + 34/2i = 0
x² - 10x + 17i = 0
For x = 5 - 3/2i:
(x - 5 + 3/2i)(x - 5 - 3/2i) = 0
x² - 5x - (3/2i)x - 5x + 25 + 15/2i + (3/2i)x - 9/2i + 3/2i² = 0
x² - 10x - 34/2i = 0
x² - 10x - 17i = 0
Now, we can compare the quadratic equation x² + 1.2x - 6.0 × 10-3 = 0 with the solved equations:
x² - 10x + 17i = 0 (Comparing coefficients)
1.2 = -10 (Comparing coefficients)
From the above comparison, we can see that the coefficient 'a' is equal to -10.