Find the domain of the function: f(x) = (x^2 - 9) / (x^2 + 5x + 6)
To find the domain of a given function, we have to find out what all values the input variable can be, in this case it is x. Given a fraction, we know that the numerator can be any value and it won't have an effect but we can't have a denominator that is equal to zero, because that will give us an undefined value. So we can say that x can be equal to any value except when it makes the denominator zero. We can now find the problem to be: for what values of x is x^2 + 5x + 6 = 0 We can do this in one of two ways: 1. We can rewrite the equation to simplify it to easily see what x could be. To simplify this specific equation, we want to find two numbers that multiply to 6 and add to 5. We can see that those numbers would be 2 and 3. So we can rewrite this equation: (x + 2) * (x + 3) = 0 Given this new format, we can see that if x is -2 or -3, our denominator will be zero. Thus our domain is all real numbers except where x is -2 or -3. 2. We can use the quadratic equation to figure out what values of x make this 0. (-b +- sqrt (b^2 - 4*a*c)) / 2a a = coefficient of x^2 b = coefficient of 5x c = 6 (-5 +- sqrt (5^2 - 4*1*6)) / (2 * 1) (-5 +- sqrt(25 - 24)) / (2) (-5 +- 1) / (2) = -6 / 2 = -3 = -4 / 2 = -2 Thus, again, we get that if x = -2 or -3, this will result in 0. So our domain is all, again, all real numbers except where x is -2 or -3.
Solve for x: 6x + 6 = -7
When solving for x, we want to get x all by itself. The first way to do this is to get 6x by itself. Since we have a +6 on the left, we want to get rid of it and we can do that by subtracting 6. Once we subtract 6 from the left, we have to do it to the right side to maintain equality. 6x + 6 = -7 - 6 -6 6x = -13 Now we have to get x by itself. Since 6x is equal to 6 * x, we have to divide by 6 to get x by itself. Again, we have to do it to both sides to isolate x. 6x = -13 ---- ---- 6 6 x = -13/6
A store has decided to reduce all their prices by 20% for the upcoming holiday. If they are selling candy for $2.50 prior to the discount, how much will the candy cost during the discount?
We know that the original price of the candy is $2.50 and that it will be reduced by 20%. Since we know that the price is going to reduce by 20%, the price will be 80% (original 100% - 20% = 80%). We can rewrite percentages in decimal form by moving the decimal place in the percent to the left by 2 digits. This way, 80% becomes 0.80 (or 0.8). We can multiply this decimal by the original price to see what the new price would become. 2.50 * 0.8, which is, $2.00