Hello, can anyone help me with this one question?
What are the two solutions of x^2 – 2x – 4 = –3x + 9?
A. the y-coordinates of the y-intercepts of the graphs of y = x^2 – 2x – 4 and y = –3x + 9
B. the x-coordinates of the x-intercepts of the graphs of y = x^2 – 2x – 4 and y = –3x + 9
C. the y-coordinates of the intersection points of the graphs of y = x^2 – 2x – 4 and y = –3x + 9
D. the x-coordinates of the intersection points of the graphs of y = x^2 – 2x – 4 and y = –3x + 9
I have no clue what the answer is. I've tried doing the problem multiple times, but I never get the same exact number on both sides.
Any help is greatly appreciated!
of course it is d.
Cool! Thank you Bob. :)
To find the solutions to the equation x^2 – 2x – 4 = –3x + 9, we need to simplify and rearrange the equation to bring all the terms to one side.
First, let's move all the terms to the left side of the equation:
x^2 – 2x – 4 + 3x - 9 = 0
Next, combine like terms:
x^2 + x - 13 = 0
Now, we have a quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = 1, and c = -13. To find the solutions, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values into the formula:
x = (-(1) ± √((1)^2 - 4(1)(-13))) / (2(1))
Simplifying:
x = (-1 ± √(1 + 52)) / 2
x = (-1 ± √53) / 2
Therefore, the two solutions to the equation x^2 – 2x – 4 = –3x + 9 are the x-coordinates of the intersection points of the graphs of y = x^2 – 2x – 4 and y = –3x + 9.
The correct answer is option D.
To find the solutions of the equation x^2 – 2x – 4 = –3x + 9, you need to solve for x.
Step 1: Start by simplifying the equation. Combine like terms on both sides of the equation:
x^2 – 2x – 4 = –3x + 9 becomes
x^2 – x – 4 = 9
Step 2: Move all terms to one side of the equation, so we have a quadratic equation equal to 0:
x^2 – x – 4 - 9 = 0
x^2 – x - 13 = 0
Step 3: Now we need to solve the quadratic equation. There are several methods to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula.
Let's use the quadratic formula in this case:
The quadratic formula states that for any quadratic equation in the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation x^2 – x - 13 = 0, we have:
a = 1, b = -1, c = -13
Plugging these values into the quadratic formula, we get:
x = (1 ± √((-1)^2 - 4(1)(-13))) / (2 * 1)
x = (1 ± √(1 + 52)) / 2
x = (1 ± √53) / 2
So the two solutions for x are (1 + √53) / 2 and (1 - √53) / 2.
Now that we have found the solutions for x, we can determine which answer choice correctly represents them.
Option A states that the solutions are the y-coordinates of the y-intercepts of the two graphs. This is not correct, as y-intercepts are the values when x = 0, and they are not relevant to finding the roots of the equation.
Option B states that the solutions are the x-coordinates of the x-intercepts of the two graphs. This is the correct answer since the x-intercepts are the points where the graphs intersect the x-axis, which are the solutions of the equation.
Option C states that the solutions are the y-coordinates of the intersection points of the two graphs. This is not correct, as we are looking for the x-coordinates of the intersection points.
Option D states that the solutions are the x-coordinates of the intersection points of the two graphs. This is also the correct answer, as mentioned earlier.
Therefore, the answer is D. the x-coordinates of the intersection points of the graphs of y = x^2 – 2x – 4 and y = –3x + 9.