Asked by Smartypants
Solve by factoring.z squared plus 4 z minus 45equals0Question content area bottomPart 1zequalsenter your response here (Use a comma to separate answers as needed.)
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Answered by
GPT-4o mini
To solve the equation \( z^2 + 4z - 45 = 0 \) by factoring, we need to find two numbers that multiply to \(-45\) (the constant term) and add to \(4\) (the coefficient of the linear term).
The numbers that satisfy these conditions are \(9\) and \(-5\) because:
- \(9 \times (-5) = -45\)
- \(9 + (-5) = 4\)
Now, we can factor the quadratic expression:
\[
z^2 + 4z - 45 = (z + 9)(z - 5) = 0
\]
Next, we set each factor equal to zero:
1. \( z + 9 = 0 \)
\( z = -9 \)
2. \( z - 5 = 0 \)
\( z = 5 \)
So the solutions to the equation \( z^2 + 4z - 45 = 0 \) are:
\[
z = -9, 5
\]
Thus, the answer is:
\(-9, 5\)
The numbers that satisfy these conditions are \(9\) and \(-5\) because:
- \(9 \times (-5) = -45\)
- \(9 + (-5) = 4\)
Now, we can factor the quadratic expression:
\[
z^2 + 4z - 45 = (z + 9)(z - 5) = 0
\]
Next, we set each factor equal to zero:
1. \( z + 9 = 0 \)
\( z = -9 \)
2. \( z - 5 = 0 \)
\( z = 5 \)
So the solutions to the equation \( z^2 + 4z - 45 = 0 \) are:
\[
z = -9, 5
\]
Thus, the answer is:
\(-9, 5\)
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