What are the two solutions of the equation 2x^2 = -x^2 – 5x – 1? A: The y-coordinates of the intersection points of the graphs y = 2x^2 and y = -x^2 - 5x - 1. B: The x-coordinates of the x-intercepts of the graph y = 2x^2 and the graph y = -x^2 - 5x - 1. C: The x-coordinates of the intersection points of the graphs y = 2x^2 and y = -x^2 - 5x - 1. D: The y-coordinates of the y-intercepts of the graphs y = 2x^2 and y = -x^2 - 5x - 1.
Mathematics · High School · Thu Feb 04 2021
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The correct answer is C: The x-coordinates of the intersection points of the graphs y = 2x^2 and y = -x^2 - 5x - 1.
To solve the equation 2x^2 = -x^2 – 5x – 1, we have to find the values of x that make the equation true. These values of x correspond to the points where the two graphs y = 2x^2 and y = -x^2 - 5x - 1 intersect. When the y-values of both graphs are equal, that means their corresponding x-values are where the graphs intersect on the x-axis. These x-values are the solutions to the equation.
Now we can rearrange the equation to find the solutions for x:
2x^2 = -x^2 – 5x – 1 Add x^2 to both sides: 2x^2 + x^2 = -5x – 1 Combine like terms: 3x^2 = -5x – 1 Add 5x and 1 to both sides: 3x^2 + 5x + 1 = 0
Now we have a standard quadratic equation, and we can find the solutions by factoring, completing the square, or using the quadratic formula. The solutions to this equation will give us the x-coordinates of the intersection points of the graphs.