A)
Multiple Choice
Solve the system of equations.
y = 2x² – 3
y = 3x – 1
A) (2, 3)
Explanation:
Since both equations are set equal to y, we can set them equal to each other and solve for x:
2x^2 - 3 = 3x - 1
2x^2 - 3x - 2 = 0
(2x + 1)(x - 2) = 0
x = -1/2 or x = 2
Now that we have the values for x, we can plug them back into one of the original equations to solve for y:
If x = -1/2:
y = 2(-1/2)^2 - 3 = 2(1/4) - 3 = 1/2 - 3 = -5/2
If x = 2:
y = 2(2)^2 - 3 = 2(4) - 3 = 8 - 3 = 5
Therefore, the solutions to the system of equations are (-1/2, -5/2) and (2, 5).
A)
How many real number solutions are there to the equation 0 = –3x² + x – 4?
(1 point)
0
1
2
3
C) 2
Explanation:
To find the number of real number solutions, we look at the discriminant of the quadratic equation:
Discriminant = b^2 - 4ac
In the equation -3x^2 + x - 4 = 0, a = -3, b = 1, c = -4.
Plugging in the values:
Discriminant = 1^2 - 4*(-3)*(-4) = 1 - 48 = -47
Since the discriminant is negative (-47), the quadratic equation has no real number solutions.
A)
Solve the equation by completing the square. If necessary, round to the nearest hundredth.
x² – 18x = 19
(1 point)
1; 19
–1; 19
3; 6
–3; 1
Completing the square for the quadratic equation x² - 18x = 19:
Step 1: Move the constant term to the other side of the equation:
x² - 18x = 19
x^2 - 18x - 19 = 0
Step 2: To complete the square, take half of the coefficient of x, square it, and add that to both sides of the equation.
Coefficient of x = -18
(-18/2)^2 = 9^2 = 81
x^2 - 18x + 81 = 19 + 81
(x - 9)^2 = 100
Step 3: Take the square root of both sides to solve for x:
x - 9 = ±10
x = 9 ± 10
This gives two possible solutions:
x = 9 + 10 = 19
x = 9 - 10 = -1
Therefore, the solutions are 19 and -1. The answer is:
1; 19
A)
Graph the quadratic functions y = -2x and y = -2x + 4 on a separate piece of paper. Using
those graphs, compare and contrast the shape and position of the graphs.
(3 points)
To graph the quadratic functions y = -2x and y = -2x + 4 separately, follow these steps:
1. For y = -2x:
- Determine points to plot. For example, choose x-values of -2, -1, 0, 1, and 2.
- Calculate corresponding y-values by substituting x into the equation.
- Plot these points and connect them smoothly to get the graph.
2. For y = -2x + 4:
- Repeat the same process by choosing x-values, calculating y-values, and plotting points for this equation.
Comparison of the graphs:
- Both functions are linear and have a negative slope (-2) indicating that they slope downwards from left to right.
- The graph of y = -2x is a line passing through the origin (0,0).
- The graph of y = -2x + 4 is also a line with the same slope but shifted upwards by 4 units.
- The graph of y = -2x + 4 intersects the y-axis at y = 4, while the graph of y = -2x passes through the origin.
Visually compare the two graphs to observe their similarities and differences in shape and position based on the steps above.