Math 11

Week 7 – Precalc 11

ximena

This week in pre-calculus 11, we learned about the discriminant. The discriminant is the radicand in the quadratic formula, b2– 4ac.

The discriminant is used to indicate whether the quadratic formula has 2 real roots or unequal roots, 1 real root or 2 equal roots, or no real roots.

x > 0: 2 real roots / 2 unequal roots.

x = 0: 1 real root / 2 equal roots.

x < 0: 0 real roots.

We use the quadratic equation to figure out which numbers are a, b, and c.

ax2 + bx + c = 0

Ex:

4x2 + 3x + 8 = 0

a = 4, b = 3, c = 8

b2– 4ac

32– 4(4)(8)

9 – 128

= -119

Seeing as the answer is negative, there are no real roots because -119 is less than 0.

Week 6 – Precalc 11

ximena

This week in Pre-calculus 11, I learned how to solve Perfect Square Trinomials.

To solve perfect square trinomials, the first and last terms in ax2 + bx + c have to be a perfect square.

Ex: x2 + 8x + 16

Then you have to factor by finding the number that multiplies twice into the last term (16), which is 4, and you have to make sure that the same number is the answer if the middle term, (a)x, is divided by 2. A different way to factor it is to square root the first and last term, and the answer for the last term square rooted should add together to equal the second/middle term.

4 x 4 = 16

8 ÷ 2 = 4

x2 + 8x + 16

= (x + 4)(x + 4)

= (x + 4)2

x = -4

To solve in the form of quadratic formula:

a = 1

b = 8

c = 16

 

*The discriminant equals 0, so we know that there can only be one real root/solution.

 

Week 5 – Precalc 11

ximena

This week in pre-calc 11, I learned how to factor polynomial expressions. C D P E U are the 5 steps to factoring a polynomial.

C = Common, D = Difference of squares, P = Pattern, E = Easy pattern, U = Ugly pattern

Common means finding a number or variable that can factor out from a binomial or a trinomial.

Ex. 64x + 16

= 16 (4x + 1)

 

Difference of squares is only used in binomials. The sign in between the expression has to be negative (-) and one number or variable has to be squared.

Ex. 25x2– 9

= (5x – 3)(5x + 3)

 

Pattern is to check if it’s a trinomial following the expression “(a)x2 + bx + c”.

– Easy patterns are when there is no coefficient in front of the x2. I then factor out to find the product of c and the sum of b with the same numbers.

Ex. x2 + 8x + 12

= (x + 6)(x+ 2)

– Ugly patterns are when there is a coefficient in front of the x2. My method is to multiply the coefficient into the term without a variable. It then becomes an easy pattern, then I factor it out. Once I get an answer with the easy pattern, I divide both sides by the coefficient of x2.

Ex. 2x2 + 3x – 2

= x2 + 3x – 4

= (x + 4)(x – 1)

= (x + \frac{4}{2})(x – \frac{1}{2})

= (x + 2)(2x – 1)

Week 4 – Precalc 11

ximena

During week 4, I learned how to add and subtract radical expressions. First, you have to make sure that all the terms are in the simplest form, then look for any like terms. If there are like terms, you add or subtract the coefficients together and leave the radicand. If there are no like terms, you cannot add or subtract them.

 

Example:

4\sqrt[]{7} + 13\sqrt[]{7}

= 17\sqrt[]{7}

 

(2\sqrt[]{5} – 6\sqrt[]{5}) + (9\sqrt[]{3} + 3\sqrt[]{3})

= -4\sqrt[]{5} + 12\sqrt[]{3}

Week 3 – Precalc 11

ximena

During week 3, I learned about the Absolute Value of a Real Number. The absolute value of a real number is the principal square root of its square or its distance from zero on a number line. The principal square root is the positive square root. The two lines “||” before and after the numbers represent absolute value.

Example: |4 – 7|  =  |-3|  =  3

*If there is a negative number outside of the “paratheses”, the answer can be negative. If there is no negative number in front of it, it has to be positive.

 

Week 2 – Precalc 11

ximena

During week 2, I learned about geometric sequences. In geometric sequences, the same number has to be multiplied into each term. The number that is being multiplied is called the common ratio. I learned how to find t_n, the ratio, or just n.

Ex. 4, 8, 16, 32, 64, …              Common Ratio = 2

Find t_{16}

t_n = a(r)^{n-1}

t_{16} = 4 (2)^{16-1}

t_{16} = 4 (32,768)

t_{16} = 131,072

Weel 1 – Precalc 11

ximena

During week 1, I learned about arithmetic sequences. In arithmetic sequences, the same number has to be added or subtracted from the terms every time. The number that is being added or subtracted is called the common difference.

 

Ex: 6, 13, 20, 27, 34, …          Common Difference = +7

Beginning at the first term (6) you add 7 every time.