Final answer:
To find the roots of the quadratic equation 9x² + 30x + 28 = 0 in simplest a+bi form, we use the quadratic formula. Since the discriminant is negative, the roots will be complex, resulting in two roots (-5/3 + √3i) and (-5/3 - √3i).
Step-by-step explanation:
The student's question involves finding the roots of a quadratic equation. To find the roots of the equation 9x² + 30x = -28, we first bring all terms to one side of the equation to get 9x² + 30x + 28 = 0. This is now in the standard form of a quadratic equation ax² + bx + c = 0.
We can solve for the roots using the quadratic formula, which is x = (-b ± √(b² - 4ac)) / (2a). Substituting the values a = 9, b = 30, and c = 28 into the formula, we get:
x = (-30 ± √((30)² - 4 × 9 × 28)) / (2 × 9)
x = (-30 ± √(900 - 1008)) / 18
x = (-30 ± √(-108)) / 18
Since the discriminant (inside the square root) is negative, this indicates that the roots will be complex. We use the property that √(-1) = i, where i is the imaginary unit. Therefore, the roots of the equation can be expressed in a + bi form as follows:
x = (-30 ± √108 × i) / 18
We can simplify √108 to 6√3, so the roots are:
x = (-30 ± 6√3i) / 18
By dividing both terms in the numerator by 18, we obtain the simplified form:
x = (-5/3 ± √3i)