Question
Which statement correctly interpret the solution to the equation
X^2 + 5x + 6 = -3x^2 - 2x
A. An input of -3 will yield an output of 0 on both sides
B. An input of -2 will yield an output of 0 on both sides
C. An input of 0 will yield an output of 0 on both sides
D. There are no input values that would yield the same output value
X^2 + 5x + 6 = -3x^2 - 2x
A. An input of -3 will yield an output of 0 on both sides
B. An input of -2 will yield an output of 0 on both sides
C. An input of 0 will yield an output of 0 on both sides
D. There are no input values that would yield the same output value
Answers
Answered by
GPT-4o mini
To interpret the solution to the equation \( X^2 + 5x + 6 = -3x^2 - 2x \), we first rearrange the equation so that one side equals zero.
Starting with:
\[ X^2 + 5x + 6 + 3x^2 + 2x = 0 \]
Combine like terms:
\[ 4x^2 + 7x + 6 = 0 \]
Next, we can determine the roots of the quadratic equation \( 4x^2 + 7x + 6 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 4 \), \( b = 7 \), and \( c = 6 \).
Calculating the discriminant:
\[ b^2 - 4ac = 7^2 - 4 \cdot 4 \cdot 6 = 49 - 96 = -47 \]
Since the discriminant is negative (\(-47\)), this indicates that there are no real solutions to the equation. Consequently, there are no input values that would yield an output of 0 on both sides of the equation.
Based on this analysis, the correct interpretation is:
**D. There are no input values that would yield the same output value.**
Starting with:
\[ X^2 + 5x + 6 + 3x^2 + 2x = 0 \]
Combine like terms:
\[ 4x^2 + 7x + 6 = 0 \]
Next, we can determine the roots of the quadratic equation \( 4x^2 + 7x + 6 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 4 \), \( b = 7 \), and \( c = 6 \).
Calculating the discriminant:
\[ b^2 - 4ac = 7^2 - 4 \cdot 4 \cdot 6 = 49 - 96 = -47 \]
Since the discriminant is negative (\(-47\)), this indicates that there are no real solutions to the equation. Consequently, there are no input values that would yield an output of 0 on both sides of the equation.
Based on this analysis, the correct interpretation is:
**D. There are no input values that would yield the same output value.**