9. x² - x - 6 (x - 3) (x + 2) Given. Remember FOIL (first, outside, inside last). In this expression, x is first, the other x is outside, the 2 is inside, and the -3 is last. If this doesn't help, just remember to multiply the numbers in the first set of parentheses by the other numbers in turn. We'll start by multiplying x by x and 2, and then move on to -3 and multiply it by the other numbers. (x) (x) = x² Because we don't know what x is, we can only define x times x as x². (x) (2) = 2x Now, we move on to x and 2. (-3) (x) = -3x Next, we're going to multiply -3 by x and 2, starting with x. (-3) (2) = -6 Now, let's add each term together. x² + 2x - 3x - 6 This is the answer we get. However, since 2x and -3x are like terms, the answer isn't simplified yet. Let's add them together and see what we get. x² - x - 6 This is the completely simplified version of the expression, since 2x - 3x = -1x, or -x. 10. y = -3/2x + 3 Let's use the points we've been given to find the answer. Remember that slope = rise/run, or y-y/x-x. (2,0) (0,3) Given 0 - 3/ 2 - 0 Plug into the slope formula. Always remember that y goes over x, and not the other way around. -3/2 Simplify. This is the slope for the line. Now, let's find the y-intercept. In the slope-intercept form of the slope formula, y = mx + b, where b is the y-intercept. (2,0) (0,3) These are our given points. If neither of them is the y-intercept, we'll have to graph the line using the slope we calculated and see where it crosses the y-axis. (0,3) Looks like the question decided to be nice and give us the y-intercept! Remember that at the y-intercept, x = 0. In this point, x = 0, so the y-intercept is the other coordinate, 3. This means that the line crosses the y-axis where y = 3. Let's plug that into the slope formula. y = -3/2x + 3 We've substituted m, or the slope, with the slope we calculated and b, or the y-intercept, with the y-intercept that we calculated. This is the equation for the line! 11. -5x² - 11x + 13 (-2x² -6x + 7) - (3x² + 5x - 6) Keep in mind that we're subtracting the first polynomial from the second polynomial, meaning that the second one comes first in the expression. -2x² - 3x² = -5x² Keep in mind that when subtracting a negative from a negative, the answer appears to be larger. For example, 1 - 1 is 0, but -1 - 1 is -2. -6x - 5x = -11x 7 + 6 = 13 We're adding 6 to 7 because subtracting a negative is the same as adding a positive. For example, if you take away someone's debt, they technically have more money than they did before. Now, let's add each term to each other and get our answer. -5x² - 11x + 13 There's our answer! Keep in mind that none of these terms go together because of the difference in variables; -5x² ends in x², 11x ends in x, and 13 has no variable. If you have an amount of apples and an amount of oranges, you can't add them together and say 'I have this many apples', because some of them are oranges. 12. -12√3 This is a simple question of factoring. There are a lot of rules to remember about factoring, though, so it can be a bit difficult to remember. -3√48 Given 48 Let's focus on the number under the radical. Now, we have to divide it with its smallest factor, other than 1, as many times as possible, and keep track of what we divide it by. 48 / 2 = 24 24 / 2 = 12 12 / 2 = 6 6 / 2 = 3 We have now divided it as many times as we can. We end up with 4 2s and a 3 (always include the final number; for instance, if you have 4 and you divide it by 2 to get 2, both 2s are considered factors). When factoring, each number has to have a pair to get outside of the radical. So, since there is only one 3, we leave it inside the radical, and take two 2s outside of the radical, since we have 4. This part can be a bit confusing, but just remember it with a fun example. You wouldn't want to go to a dance without a partner, right? No factors can go outside of the radical without a partner, and when they do, they're considered one (in this case, when we have 4 2s, each pair of 2 is considered one, so we get 2 2s). When you go to a dance with a partner, you're considered one couple, and that poor lonely 3 is stuck inside his house with no one to go out with. Now we have 2 2s outside the radical, and the -3 that was already outside. Let's multiply each number together to get the number outside the radical. 2 x 2 x -3 = -12 The number outside the radical is -12. The number inside is lonely 3, as we determined earlier. So, the answer is -12√3. Hope this helps!

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