Question
Find all solutions for 4x^2 + 32x + 60 = 0
Answers
Answered by
GPT-4o mini
To solve the quadratic equation \(4x^2 + 32x + 60 = 0\), we can start by simplifying the equation. First, we notice that each term can be divided by 4:
\[
x^2 + 8x + 15 = 0
\]
Next, we can factor the quadratic. We need to find two numbers that multiply to \(15\) (the constant term) and add up to \(8\) (the coefficient of \(x\)). The numbers \(3\) and \(5\) satisfy these conditions, since \(3 \times 5 = 15\) and \(3 + 5 = 8\).
Thus, we can factor the equation as follows:
\[
(x + 3)(x + 5) = 0
\]
Now, we can set each factor equal to zero:
1. \(x + 3 = 0\) leads to \(x = -3\)
2. \(x + 5 = 0\) leads to \(x = -5\)
Thus, the solutions to the equation \(4x^2 + 32x + 60 = 0\) are:
\[
\boxed{-3} \quad \text{and} \quad \boxed{-5}
\]
\[
x^2 + 8x + 15 = 0
\]
Next, we can factor the quadratic. We need to find two numbers that multiply to \(15\) (the constant term) and add up to \(8\) (the coefficient of \(x\)). The numbers \(3\) and \(5\) satisfy these conditions, since \(3 \times 5 = 15\) and \(3 + 5 = 8\).
Thus, we can factor the equation as follows:
\[
(x + 3)(x + 5) = 0
\]
Now, we can set each factor equal to zero:
1. \(x + 3 = 0\) leads to \(x = -3\)
2. \(x + 5 = 0\) leads to \(x = -5\)
Thus, the solutions to the equation \(4x^2 + 32x + 60 = 0\) are:
\[
\boxed{-3} \quad \text{and} \quad \boxed{-5}
\]