A. Solve the following quadratic equations. Make sure to show all your work. Do not use any method (e.g. factoring, completing the square, quadratic formula, graphing) more than twice. Use the graphing method at least once.

1. 3x+2 + 11x -20 =0
2. x2 + 3x -4 =0
3. 3x2 - x - 1 =0

B. State the different methods you used to solve each equation. Make sure that your work demonstrates both algebraic and graphing methods.

I am so lost on this! Could someone please show me how all of this is done? It will probably be on my tests!

Thanks much!

Sure! Let's go step by step and solve each of the quadratic equations using different methods.

1. 3x + 2 + 11x - 20 = 0

Start by combining like terms:
14x -18 = 0

Now, to solve this equation using the graphing method, we first need to plot the graph of the equation y = 14x -18. This can be done by choosing different x-values and calculating the corresponding y-values. For simplicity, let's use x-values -5, -4, -3, ..., 4, 5.

x = -5 → y = 14(-5) - 18 = -98
x = -4 → y = 14(-4) - 18 = -82
x = -3 → y = 14(-3) - 18 = -66
x = -2 → y = 14(-2) - 18 = -50
x = -1 → y = 14(-1) - 18 = -34
x = 0 → y = 14(0) - 18 = -18
x = 1 → y = 14(1) - 18 = -4
x = 2 → y = 14(2) - 18 = 10
x = 3 → y = 14(3) - 18 = 26
x = 4 → y = 14(4) - 18 = 42
x = 5 → y = 14(5) - 18 = 58

Plotting these points on a graph and connecting them with a line, we can find the x-values where the line intersects the x-axis. These are the solutions to the equation.

Analytically, to solve the equation algebraically, we continue as follows:

14x - 18 = 0
14x = 18
x = 18/14
x = 9/7

Therefore, the solution to the equation is x = 9/7.

2. x^2 + 3x - 4 = 0

This equation can be solved using factoring.

x^2 + 3x - 4 = 0
(x + 4)(x - 1) = 0

Setting each factor equal to zero and solving for x, we have:

x + 4 = 0 -> x = -4
x - 1 = 0 -> x = 1

The solutions to the equation are x = -4 and x = 1.

3. 3x^2 - x - 1 = 0

This equation can be solved using the quadratic formula.

The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values from the equation into the formula, we have:

x = (-(-1) ± √((-1)^2 - 4(3)(-1))) / (2(3))
x = (1 ± √(1 + 12)) / 6
x = (1 ± √13) / 6

Therefore, the solutions to the equation are x = (1 + √13) / 6 and x = (1 - √13) / 6.

In summary, for each equation:
1. We used the graphing method to find the solution x = 9/7.
2. We used the factoring method to find the solutions x = -4 and x = 1.
3. We used the quadratic formula to find the solutions x = (1 + √13) / 6 and x = (1 - √13) / 6.