Help with any of these would be greatly appreciated! THANK YOU!
2. How would i determine the values of 'k' if the graph y=2x^2-2x+3k intersects the x-axis at two distinct points?
3. The path of a basketball can be modeled by the equation h(d)=-0.13d^2+d+2. where h(d) is the height in meters or the basketball and d is the horizontal distance in meters from the player. How far does the ball travel horizontally before hitting the floor?
4. How would i solve x^3-3x^2+2=0?
How would i show it in exact values?
6. How would i solve |x| (greater than or equal to) |x - 3| using my graphing calc. What are the steps?
7. What would be the diameter of the largest circle possible that could be drawn between the parallel lines of y = 2x and y = 2x - 5?
2. This has already been answered, see previous post.
3. This has already been answered, see previous post.
2. To determine the values of 'k' for which the graph intersects the x-axis at two distinct points, you need to find the discriminant of the quadratic equation. The discriminant is calculated as b^2 - 4ac, where the equation is in the form ax^2 + bx + c = 0. In this case, your equation is y = 2x^2 - 2x + 3k. So, the discriminant can be expressed as (-2)^2 - 4(2)(3k). The discriminant needs to be greater than 0 for the graph to intersect the x-axis at two distinct points. Therefore, you need to solve the inequality:
(-2)^2 - 4(2)(3k) > 0
Simplifying further, you get:
4 - 24k > 0
Now, solve for 'k':
-24k > -4
k < 1/6
So, any value of 'k' that is less than 1/6 will make the graph intersect the x-axis at two distinct points.
3. To find how far the basketball travels horizontally before hitting the floor, you need to determine when the height h(d) reaches zero. Set h(d) = 0 and solve for 'd':
-0.13d^2 + d + 2 = 0
To solve this quadratic equation, you can use the quadratic formula:
d = [-b ± √(b^2 - 4ac)] / (2a)
In this case, a = -0.13, b = 1, and c = 2. Plug in the values and solve for 'd':
d = [-(1) ± √((1)^2 - 4(-0.13)(2))] / (2(-0.13))
After evaluating the equation, you will get two values for 'd'. The positive value represents the horizontal distance traveled before hitting the floor, as distance cannot be negative.
4. To solve the equation x^3 - 3x^2 + 2 = 0, you can apply the Rational Root Theorem and synthetic division technique. First, list all the possible rational roots of the equation by considering the factors of the constant term (2) divided by the factors of the leading coefficient (1). In this case, the possible rational roots are ± 1 and ± 2. Perform synthetic division to test each root and find the real roots of the equation. By testing these values, you will find that 1 is a root. Divide the equation by (x - 1) to obtain the remaining quadratic equation x^2 - 2x - 2 = 0. By applying the quadratic formula or factoring, you can find the remaining roots of the equation. Therefore, the exact values of the solution are x = 1, x = 1 + √3, and x = 1 - √3.
6. To solve the inequality |x| ≥ |x - 3| using a graphing calculator, follow these steps:
Step 1: Graph y = |x|.
Step 2: Graph y = |x - 3|.
Step 3: Identify the points of intersection on the graph.
Step 4: Determine the region on the x-axis where |x| is greater than or equal to |x - 3|. This region will be to the left of the leftmost point of intersection and to the right of the rightmost point of intersection.
7. The parallel lines y = 2x and y = 2x - 5 have the same slope of 2. The distance between parallel lines is equal to the perpendicular distance between any point on one line and the other line. This distance can be found using the formula for the distance between a point and a line. The formula is:
Distance = |Ax + By + C| / sqrt(A^2 + B^2)
In this case, A = 2, B = -1 (the coefficients of x and y in the equation y = 2x - 5), and C = 5. Plug in these values into the formula:
Distance = |2x + (-1)y + 5| / sqrt((2)^2 + (-1)^2)
Simplifying further, you get:
Distance = |2x - y + 5| / sqrt(5)
Therefore, the diameter of the largest circle that can be drawn between the parallel lines is twice the distance, which is:
Diameter = 2 * Distance = 2 * |2x - y + 5| / sqrt(5)