4.5+&+4.6

= 10 9 8 7 6 5 4 3 2 1 BLAST OFF!!! =

** Vocabulary 4.5 **
Square Root - a number that produces a specified quantity when multiplied by itself

Radical - of or going to the root or origin

Radicand - the number or expression beneath a radical sign

Rationalizing the Denominator - the process of eliminating a radical expression in the denominator of a fraction by multiplying both the numerator and denominator by an appropriate radical expression

1. For an explanation of solving quadratic equations by finding square roots :
media type="custom" key="23791038"

media type="custom" key="23791082"

**2**. Solve x2 – 4 = 0.

Previously, I'd solved this by factoring the difference of squares, and solving each factor; the solution was "x = ± 2". However—

I can also try isolating the squared variable term, putting the number over on the other side, like this:

x2 – 4 = 0

x2 = 4

I know that, when solving an equation, I can do whatever I like to that equation as long as I do the same thing to both sides of the equation. On the left-hand side of this particular equation, I have an x2, and I need a plain x. To turn an x2 into an x, I can take the square root of each side of the equation:

x = ± 2

Then the solution is x = ± 2

Why did I need the "±" ("plus-minus") sign on the 2 when I took the square root of the 4? Because it might have been a positive 2 or a negative 2 that was squared to get that 4 in the original equation. == ==


 * Vocabulary 4.6 **

Imaginary Unit i - The square root of -1, corresponding to the point (0,1) in the geometric representation of complex numbers as points in a plane.

Complex Number - a number of the form a+bi where a and b are real numbers and i is the square root of -1

Imaginary Number - any complex number of the form i b, where i = √--1

Complex Conjugates - the complex number whose imaginary part is the negative of that of a given complex number, the real parts of both numbers being equal: a--ibisthecomplexconjugateofa+ ib

Complex Plane - a plane the points of which are complex numbers.


 * __ Example __** :

The complex conjugate of 3 – 4//i// is 3 + 4//i.// Follow these steps to finish the problem:

> > You go with (1 + 2i)(3 + 4i) = 3 + 4i + 6i + 8i2, which simplifies to (3 – 8) + (4i + 6i), or –5 + 10i. > You have (3 – 4i)(3 + 4i), which FOILs to 9 + 12i – 12i – 16i2. Because i2 = –1 and 12i – 12i = 0, you're left with the real number 9 + 16 = 25 in the denominator (which is why you multiply by 3 + 4i in the first place). > This answer still isn't in the right form for a complex number, however. > Notice that the answer is finally in the form A + Bi.
 * 1) Multiply the numerator and the denominator by the conjugate.
 * 1) FOIL the numerator.
 * 1) FOIL the denominator.
 * 1) Rewrite the numerator and the denominator.
 * 1) Separate and divide both parts by the constant denominator.

How to solve equations with complex numbers : media type="custom" key="23791048"

media type="custom" key="23791084"