Question
Select all the solutions to the equation y=3x^2+18x+24
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Answers
GPT-4o mini
To solve the equation \( y = 3x^2 + 18x + 24 \) for \( y = 0 \), we need to set the equation equal to zero:
\[
3x^2 + 18x + 24 = 0
\]
Next, we can simplify this quadratic equation by dividing all terms by 3:
\[
x^2 + 6x + 8 = 0
\]
Now, we can factor the quadratic equation:
\[
(x + 2)(x + 4) = 0
\]
Setting each factor to zero gives us:
1. \( x + 2 = 0 \) which leads to \( x = -2 \)
2. \( x + 4 = 0 \) which leads to \( x = -4 \)
Thus, the solutions to the equation \( y = 3x^2 + 18x + 24 \) when \( y = 0 \) are:
\[
x = -2 \quad \text{and} \quad x = -4
\]
If you need the corresponding \( y \) values for those \( x \) values, they would both be \( y = 0 \) since we set \( y \) to 0 in the beginning.
So, the complete set of solutions is \( (-2, 0) \) and \( (-4, 0) \).
\[
3x^2 + 18x + 24 = 0
\]
Next, we can simplify this quadratic equation by dividing all terms by 3:
\[
x^2 + 6x + 8 = 0
\]
Now, we can factor the quadratic equation:
\[
(x + 2)(x + 4) = 0
\]
Setting each factor to zero gives us:
1. \( x + 2 = 0 \) which leads to \( x = -2 \)
2. \( x + 4 = 0 \) which leads to \( x = -4 \)
Thus, the solutions to the equation \( y = 3x^2 + 18x + 24 \) when \( y = 0 \) are:
\[
x = -2 \quad \text{and} \quad x = -4
\]
If you need the corresponding \( y \) values for those \( x \) values, they would both be \( y = 0 \) since we set \( y \) to 0 in the beginning.
So, the complete set of solutions is \( (-2, 0) \) and \( (-4, 0) \).