This week in pre-calc, we learned how to solve a quadratic equation by factoring. The equation + bx + c = 0. A, B, & C are all constant terms and A cannot be equal to 0. When factoring a quadratic equation, you use the zero product property. This means you take your quadratic equation and you start by factoring. Once you’re done factoring, you’ll have 2 terms that should be equal to 0.
Ex. -5x + 6 = 0
(x – 2)(x-3) = 0
Once you factor your equation, you take each term and find out the value of x in order for the term to be equal to 0.
Ex. (x-2) = a (x-3) = b
a = (x-2) = 0, x = +2
b = (x-3)=0, x = +3
Basically you’re taking the opposite number to find x in order to make the term equal to 0.
(x – 2)(x-3) = 0
(2-2)(3-3) = 0
This week in pre-calc 11, we learned how to factor polynomial expressions. There are 5 steps you can use to factor a polynomial, CDPEU. CDPEU stands for Common, Difference of squares, Pattern, Easy pattern and Ugly pattern.
Common means finding any like terms in the expression and combining them together
Difference of squares is used in binomials only (2 term expression) and there must be a — sign because it’s a difference. Difference of squares only works if everything is a perfect square. When factoring a difference of squares, they will always be conjugates so there is no middle term
64 = 8x x 8x
25 = 5×5
(8x+5)(8x-5) = 64
Pattern, Easy pattern and Ugly patterns are used for trinomials only (expression with 3 terms)
Pattern is figuring out if the expression is an easy pattern or an ugly pattern.
Easy pattern +6x+8 it’s simple because there is no coefficient in the first term and you find 2 numbers that add to the middle term and multiply to the term.
8 = 1×8, 2×4 2+4 = 6
(x+2)(x+4) = +6x+8
Ugly pattern is +11x-10 an ugly pattern is ugly because there is a coefficient with the first term. An easy way to factor an ugly pattern is using a box
Some expressions do not factor.
This week in Pre-Calc, we learned how to add and subtract radical expressions. It’s useful being able to add and subtract radicals together to make the expression look a lot easier and you’re more organized. To add or subtract radicals, you need to simplify as low as possible so that the radicand cannot be a perfect square.
When you simplify your radicals down, you can only add common radicals together. When you’re adding or subtracting radicals, the radicand always stays the same, which is why you need to have a common radicand. Once you’ve collected common radicals you you either add or subtract the co-efficients together depending on the sign.
This week in Pre-Calc, we learned about finding the absolute value of a real number. The absolute value of a real number is the principal square root of the square of a number. To the find absolute value of a number, you just count how far away the number is from 0.
Ex. 14 = 14
14 is only 14 away from 0 so the absolute value is 0
Ex. -14 = 14
-14 is also 14 away from 0 so the absolute value is 0
When you’re finding the absolute value in between the long brackets, it will always be a positive outcome, even for negative numbers.
This week in pre-calc 11, I learned how to find the finite sum of a converging geometric sequence. A converging geometric sequence is where the number keeps getting smaller and smaller but will never equal to 0. To figure out of a geometric sequence is converging, there are 2 ways you can tell, 0 < r < 1 and -1 < r < 0.
To figure out a geometric sum, you need to find the common ratio and use the formula and replace ‘a’ with the first term and replace ‘r’ with the common ratio. It’s important to know if its converging because then you know there is no finite ending to the sequence.
4, 8, 12, 16, 20…
General Equation – = + ()d
d = 4
= 4 + 4
= 4 + (2)4
= 4 + (49)4
= 25 (204)
This week in Pre-Calc, we learned about finding the sum of an arithmetic series. An arithmetic series is the sum of all the terms in an arithmetic sequence. Using the formula makes it easier to find the sum without having to add up all the numbers 1 by 1.
To find the sum of an arithmetic series, you use the formula . N represents the amount of terms you have in your sequence . You replace with the first term in your sequence and with the last term in your sequence. When you have your formula ready, you add your first term and last term together and you divide ‘n’ by 2. Because you add the first and last term together, every other term going in the opposite direction should add to the same amount as the first and last.
Ex. 1 + 10 = 11, 2 + 9 = 11, 3 + 8 = 11
Because you’re adding the terms together, you divide the amount of terms you have in half and then multiply by the first and last term added together.
Ex. 10 ÷ 2 = 5, 5 (11) = 55
The total of the sequence would be 55