April's Responses in WyzAnt Answers

The wording on the question is a little confusing, but I think I get what you're saying. Someone has thrown 50 darts at a dart board where a bullseye is worth 7 points and anything else is worth 2 points. The person has 152 points, and the question is to find out how many bullseyes were made. 

I think the numbers here are a bit off, which might be why you're having trouble. I'll walk through how I would solve the problem using equations and explain.

There are 2 types of results from throwing a dart, bullseye or not. We don't know how many of either, so we'll let b=number of bullseyes and n = number of non-bullseyes.

Since there are 50 darts, we know that: b + n = 50

The total number of points is 152: bullseye points + non-bullseye points. All the bullseye points would be 7*b (7b) and non-bullseye points would be 2*n (2n). So we could also write 7b + 2n = 152

Now we have the system of equations
  b + n   = 50
7b + 2n = 152

I'm going to use substitution to solve this problem. Look at the first equation: if we subtract b from both sides, we see that n = 50-b. Substitute (50-b) in place of n in the 2nd equation.

7b + 2(50-b) = 152
7b + 100 - 2b = 152 (distribute)
5b +100 = 152 (combine like terms)
5b = 52 (subtract 100 from both sides)
b = 10.4 (divide both sides by 5)

Does this answer make sense? Can you have 10.4 bullseyes? Nope! Now, if there were 51 darts thrown instead, everything works out rather nicely. 

Great question!  When you are graphing compound inequalities, it's very important to note whether the word joining them is "AND" or "OR".  When using "or", the solutions are any values that satisfy AT LEAST ONE inequality. Sometimes this looks like two arrows going separate directions. With "and", any solutions must satisfy BOTH inequalities.  Sometimes this looks like a line segment.

In your example x<4 and x>6, you want solutions that are both less than 4 and greater than 6.  But no such numbers exist!  If you had x<4 and x<6, then your solution would be all numbers less than 4 (since all numbers less than 4 are also less than 6).  However, in your example, there are no such numbers that satisfy both.  Therefore, there is no solution.  On a number line, this looks like an empty number line.

Let me know if you need clarification!  It's easier to visualize with drawing than text, but I hope this was enough of an explanation with just words.  :)

Remember that the roots satisfy f(x)=0.  Some time ago you solved polynomials by factoring, and used the zero-product property to find the roots.  For example, suppose you factored x²-4x-5=0 and got (x+1)(x-5)=0.  By the zero-product property, your roots are -1 and 5.

Going from the roots to the polynomial is like going backwards.  Suppose you are given -2, 3, and 0 as roots.  You could write it in the factored form:

x(x+2)(x-3)=0

(You can check by plugging in the roots and seeing that the result=0.) 

Now to find the polynomial, multiply the terms.

x(x+2)(x-3) = (x²+2x)(x-3) = x³-3x²+2x²-6x = x³-x²-6x  
So in my sample, x³-x²-6x is the polynomial with roots -2, 0, and 3

I hope this answers your question!  Let me know if you need clarification.

Hi there. Like terms are terms that have the same variable factors. 2x and 2y are not like terms, but 5x and -8x are like terms. x² and x are NOT like terms, because x²=x*x (it has two variable factors). In an expression, find the like terms and add their coefficients to combine like terms.

5x²-6-12x²+12+2x

-7x² + 2x + 6

Don't add exponents. Remember that x*x = x² but 2x = x + x.

When working to try to get a variable by itself, you perform the inverse operation. The 15 is being subtracted from -3y, so the opposite is adding 15 to both sides.

3 +15 = -3y -15 +15

18 =-3y