Question

select the correct answer.
one solution to a quadratic function, g, is given.
$9 - \sqrt{2}i$
which statement is true?
a. function g has no other solutions.
b. the other solution to function g is $-9 + \sqrt{2}i$
c. the other solution to function g is $-9 - \sqrt{2}i$
d. the other solution to function g is $9 + \sqrt{2}i$

Explanation:

Step1: Recall Complex Conjugate Root Theorem

For a quadratic function with real coefficients, if a complex number \(a + bi\) is a root, then its complex conjugate \(a - bi\) is also a root. Here, the given root is \(9 - \sqrt{2}i\), where \(a = 9\) and \(b=\sqrt{2}\).

Step2: Find the Conjugate Root

The complex conjugate of \(9 - \sqrt{2}i\) is \(9 + \sqrt{2}i\) (we change the sign of the imaginary part). So the other solution should be \(9 + \sqrt{2}i\).

Step3: Analyze Options

• Option A: Quadratic functions (degree 2) have two roots (real or complex), so A is false.
• Option B: \(-9 + \sqrt{2}i\) is not the conjugate of \(9 - \sqrt{2}i\), so B is false.
• Option C: \(-9 - \sqrt{2}i\) is not the conjugate of \(9 - \sqrt{2}i\), so C is false.
• Option D: \(9 + \sqrt{2}i\) is the conjugate of \(9 - \sqrt{2}i\), so D is true.

Answer:

D. The other solution to function \( g \) is \( 9 + \sqrt{2}i \)