Question
One solution of a quadratic equation is 3+5i . a) What is the other solution?
Answers
Answer
b) Explain how you know.
Answered by
GPT-4o mini
For a quadratic equation with real coefficients, if one solution is a complex number, the other solution must be its conjugate.
Given that one solution is \( 3 + 5i \), the other solution will be the conjugate of this number, which is:
\[
3 - 5i
\]
So, the other solution is \( 3 - 5i \).
Given that one solution is \( 3 + 5i \), the other solution will be the conjugate of this number, which is:
\[
3 - 5i
\]
So, the other solution is \( 3 - 5i \).
Answered by
GPT-4o mini
The reason we know that the other solution of the quadratic equation must be the conjugate is due to the properties of polynomial equations with real coefficients.
1. **Complex Roots Theorem**: If a polynomial has real coefficients, any non-real roots must occur in conjugate pairs. This means if \( a + bi \) is a root (where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit), then \( a - bi \) must also be a root.
2. **Quadratic Equations**: A quadratic equation can be expressed in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are real numbers. If we find a complex root such as \( 3 + 5i \), its conjugate \( 3 - 5i \) must also satisfy the equation due to this theorem.
3. **Effects on the Polynomial**: The presence of a complex number as a root indicates specific symmetrical properties in the polynomial's graph. These properties ensure that the corresponding conjugate must also be a root; their combined influence guarantees that the coefficients (which are real) of the polynomial remain unchanged.
Therefore, since \( 3 + 5i \) is given as a solution to the quadratic equation, we can conclude that the other solution is \( 3 - 5i \) due to the requirement that non-real solutions appear in conjugate pairs when the coefficients are real.
1. **Complex Roots Theorem**: If a polynomial has real coefficients, any non-real roots must occur in conjugate pairs. This means if \( a + bi \) is a root (where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit), then \( a - bi \) must also be a root.
2. **Quadratic Equations**: A quadratic equation can be expressed in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are real numbers. If we find a complex root such as \( 3 + 5i \), its conjugate \( 3 - 5i \) must also satisfy the equation due to this theorem.
3. **Effects on the Polynomial**: The presence of a complex number as a root indicates specific symmetrical properties in the polynomial's graph. These properties ensure that the corresponding conjugate must also be a root; their combined influence guarantees that the coefficients (which are real) of the polynomial remain unchanged.
Therefore, since \( 3 + 5i \) is given as a solution to the quadratic equation, we can conclude that the other solution is \( 3 - 5i \) due to the requirement that non-real solutions appear in conjugate pairs when the coefficients are real.
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