- Although sometimes, if there is 2 terms in the denominator, we can multiply it by its **conjugate**. Recall **difference of squares**, $`(a + b) \text{ and } (a-b)`$ are **conjugates** of one another.
- Then, the denominator becames a difference of squares, and we got rid of the radical.
- The **standard form** of a quadraic is $`ax^2 + bx + c = 0`$, where $`a`$ is the **quadratic coefficient**, $`b`$ is the **linear coefficient**, and $`c`$ is the **constant coefficient**.
- You can solve a quadratic by factoring/decomposition, then applying the **Zero Factor Principle**, and solve for $`x`$. The **Zero Factor Principle** is if $`A \times B = 0`$, then either $`A = 0`$ or $`B = 0`$.
## Completing The Square
- This process is simply trying to create a perfect trinomial, while still balancing the equation/making the equation true
- The **Standard form** of a quadratic function $`y = ax^2+bx+c`$ can be rearranged to **Vertex form**, $`y = a(x-h)^2 + k`$ through completing the square, the **Vertex** $`(h, k)`$ can be eaisly read from this form.
### Steps To Complete The Square
- Factor out the $`a`$ coefficient from the first 2 terms. Make sure to put brackets around them.
- Add and subtract within the brackets $`(\dfrac{b}{2a})^2`$
- Remove the bracket from step 1 by applying **distrubutive property** (multiplying $`a`$/the **quadratic coefficient**)
- Factor the **perfect trinomial** that was created, and combine **like terms**.
### Solving Quadratic Equations By Completing The Square
- First complete the square of the quadratic equation/function.
- Move the constant terms to the other side.
- Square both sides.
- Isolate $`x`$
## Quadratic Formula
```math
x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\
\text{Where } ax^2+ bx + c = 0, a =\not 0, \text{ and } x \text{ are the roots of that quadratic equation}
```
- The formula is derived from completing the square.
- The **sums of the roots** is simply $`\dfrac{-b}{a}`$, or $`X_1 + X_2 = \dfrac{-b}{a}`$
- The **products of the roots**, is simply $`\dfrac{c}{a}`$, or $`(X_1)(X_2) = \dfrac{c}{a}`$
- The **Axis of Symmetry** is at $`\drac{-b}{2a}`$
## Discriminant
- $`D =b^2 - 4ac`$, this is also part of the **quadratic formula**!
- If $`D > 0`$, the quadratic equation has 2 distinct real roots
- If $`D = 0`$, the quadratic equation has 1 distinct real root or 2 equal real roots
- If $`D <0`$,thequadraticequationhasnorealroots
|Two distinct real roots|One real root|No real roots|
- $`i = \sqrt{-1}`$. This equation has no solution in the set of real numbers
- An expression in the from $`a + bi`$, called the rectangular from, where $`a`$ and $`b`$ are real numbers, and $`i`$ is a complex number.
- The set of complex numbers includes the real numbers since any real number $`x`$ can be written as $`x + i(0)`$.
- $`a+bi`$ and $`a-bi`$ are conjugates(same term with opposite signs).
- Complex roots of a quadratic quation occurs in **conjugate pairs**, recall discriminant, if its less than 0, there are 2 complex roots that are **conjugates** ($`a \pm bi`$)
