Simplify the expression.
(–6 + i) over (–5 + i)
(1 point)
Responses
(31 + i) over 26
Image with alt text: (31 + i) over 26
(29 + i) over 26
Image with alt text: (29 + i) over 26
(31 + i) over 24
Image with alt text: (31 + i) over 24
(31 + 11i) over 26
To solve the expression (–6 + i) over (–5 + i), we can multiply both the numerator and denominator by the conjugate of the denominator to eliminate the imaginary terms.
The conjugate of –5 + i is –5 - i.
So, we have:
(–6 + i)(–5 - i) / (–5 + i)(–5 - i)
Expanding the numerator and the denominator:
(–6)(–5) + (–6)(–i) + (i)(–5) + (i)(–i) / (–5)(–5) + (–5)(–i) + (i)(–5) + (i)(–i)
Calculating the expanded terms:
30 + 6i - 5i - i^2 / 25 + 5i - 5i - i^2
Since i^2 is equal to -1, we can simplify further:
30 + i + 1 / 25 - 1
Simplifying the numerator and denominator:
31 + i / 24
Therefore, the simplified expression is (31 + i) over 24.
To simplify the expression (–6 + i) over (–5 + i), we need to rationalize the denominator.
To do this, we multiply both the numerator and the denominator by the conjugate of the denominator, which is (–5 – i).
(–6 + i) / (–5 + i) * (–5 – i) / (–5 – i)
Simplifying:
(-6 + i)(-5 - i) / (–5 + i)(–5 – i)
Expanding the numerator and the denominator:
(30 + 6i + 5i - i^2) / (25 + 5i - 5i - i^2)
Simplifying further:
(30 + 11i + 1) / (25 + 1)
(31 + 11i) / 26
Therefore, the simplified expression is (31 + 11i) over 26.
To simplify the expression (–6 + i) over (–5 + i), we can multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of –5 + i is –5 – i.
So, the expression becomes:
(–6 + i) * (–5 – i) over (–5 + i) * (–5 – i)
Expanding the numerator and the denominator, we get:
(-6 * -5) + (-6 * -i) + (i * -5) + (i * -i) over (-5 * -5) + (-5 * -i) + (i * -5) + (i * -i)
Simplifying further, we get:
30 + 6i - 5i - i^2 over 25 + 5i - 5i - i^2
Since i^2 is equal to -1, we can substitute it in the equation:
30 + i(6 - 5) - (-1) over 25 - (-1)
Simplifying the terms inside the parentheses, we get:
30 + i(1) + 1 over 25 + 1
Adding the like terms, we get:
31 + i over 26
Therefore, the simplified expression is (31 + i) over 26.