Question
Rewrite the following equation in a quadratic equation and solve by factoring: d^2 + 3x = 6x + 10. Which of the following contains the correct values for x?
1. 2, 5
2. -2, -5
3. 2, -5
4. -2, 5
1. 2, 5
2. -2, -5
3. 2, -5
4. -2, 5
Answers
Answered by
GPT-4o mini
To rewrite the equation \( d^2 + 3x = 6x + 10 \) in standard quadratic form, let's first rearrange it. We'll move all terms to one side of the equation:
\[
d^2 + 3x - 6x - 10 = 0
\]
This simplifies to:
\[
d^2 - 3x - 10 = 0
\]
Now, we'll factor this equation. We can treat this as a quadratic in terms of \( x \):
\[
d^2 = 3x + 10
\]
To rewrite this as a standard form quadratic equation in \( x \), we would typically set it to zero:
\[
-3x + d^2 - 10 = 0
\]
Rearranging gives us:
\[
3x = d^2 - 10 \Rightarrow 3x - d^2 + 10 = 0
\]
Reorganizing shows that we can isolate \( x \) leading to:
\[
x = \frac{d^2 + 10}{3}
\]
Now, to find the integer values of \( x \) and whether they fit with the provided selections (2, 5), (-2, -5), (2, -5), (-2, 5), we need to understand which values of \( d \) would result in those \( x \) values.
1. **When \( x = 2 \)**:
\[
3(2) = d^2 + 10 \implies 6 = d^2 + 10 \implies d^2 = -4 \quad \text{(not valid)}
\]
2. **When \( x = 5 \)**:
\[
3(5) = d^2 + 10 \implies 15 = d^2 + 10 \implies d^2 = 5 \quad \text{(valid, } d = \sqrt{5} \text{)}
\]
3. **When \( x = -2 \)**:
\[
3(-2) = d^2 + 10 \implies -6 = d^2 + 10 \implies d^2 = -16 \quad \text{(not valid)}
\]
4. **When \( x = -5 \)**:
\[
3(-5) = d^2 + 10 \implies -15 = d^2 + 10 \implies d^2 = -25 \quad \text{(not valid)}
\]
Thus, the only valid result for integer \( x \) from this scenario comes from \( x = 5 \). None of the other conditions yield feasible matches.
Hence, there are no valid values for \( x \) from the provided options. However, if it were necessary to examine potential implications, values leading up to results for other \( d \) variations, we realize \( x = 5 \) yields solutions. Therefore, based on these exploration contexts, the provided options are limited, suggesting focus on the alternatives with regard to \( d \).
Regarding basic identification:
The correct values seem to be **none of the above options hold since all scenarios lead to unusable \( d^2 \)**, while counting for integers solidifying necessary investigations on the alternative confirmations should coincide into a definitive value under the generated simplifications extended here.
\[
d^2 + 3x - 6x - 10 = 0
\]
This simplifies to:
\[
d^2 - 3x - 10 = 0
\]
Now, we'll factor this equation. We can treat this as a quadratic in terms of \( x \):
\[
d^2 = 3x + 10
\]
To rewrite this as a standard form quadratic equation in \( x \), we would typically set it to zero:
\[
-3x + d^2 - 10 = 0
\]
Rearranging gives us:
\[
3x = d^2 - 10 \Rightarrow 3x - d^2 + 10 = 0
\]
Reorganizing shows that we can isolate \( x \) leading to:
\[
x = \frac{d^2 + 10}{3}
\]
Now, to find the integer values of \( x \) and whether they fit with the provided selections (2, 5), (-2, -5), (2, -5), (-2, 5), we need to understand which values of \( d \) would result in those \( x \) values.
1. **When \( x = 2 \)**:
\[
3(2) = d^2 + 10 \implies 6 = d^2 + 10 \implies d^2 = -4 \quad \text{(not valid)}
\]
2. **When \( x = 5 \)**:
\[
3(5) = d^2 + 10 \implies 15 = d^2 + 10 \implies d^2 = 5 \quad \text{(valid, } d = \sqrt{5} \text{)}
\]
3. **When \( x = -2 \)**:
\[
3(-2) = d^2 + 10 \implies -6 = d^2 + 10 \implies d^2 = -16 \quad \text{(not valid)}
\]
4. **When \( x = -5 \)**:
\[
3(-5) = d^2 + 10 \implies -15 = d^2 + 10 \implies d^2 = -25 \quad \text{(not valid)}
\]
Thus, the only valid result for integer \( x \) from this scenario comes from \( x = 5 \). None of the other conditions yield feasible matches.
Hence, there are no valid values for \( x \) from the provided options. However, if it were necessary to examine potential implications, values leading up to results for other \( d \) variations, we realize \( x = 5 \) yields solutions. Therefore, based on these exploration contexts, the provided options are limited, suggesting focus on the alternatives with regard to \( d \).
Regarding basic identification:
The correct values seem to be **none of the above options hold since all scenarios lead to unusable \( d^2 \)**, while counting for integers solidifying necessary investigations on the alternative confirmations should coincide into a definitive value under the generated simplifications extended here.
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