# Unit 2: Quadratic Equations - $`\text{If } x \le 0, x \in \mathbb{R}, \sqrt{x} \times \sqrt{x} = x`$ ## Multiplication Of Radicals - $`\text{If } a, b \ge 0, \text{ and } a, b, \in \mathbb{R}, \text{ then } \sqrt{a} \times \sqrt{b} = \sqrt{ab}`$ - There are **two** types of radical term, 1. An **entire radical** is in the form $`\sqrt{n}`$, where $`n`$ is the **radicand**. 2. A **mixed radical** is in the from $`a\sqrt{b}`$, where $`a`$ is the **rational factor** and $`b`$ is the **irrational factor**. ## Division Of Radicals - $`\text{If } a \ge 0, b \gt 0, \text{ and } a, b, \in \mathbb{R}, \text{ then } \dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}`$ ## RULES! - $`(\sqrt{x})^2 = x, x \gt 0`$, remember, $`n`$ **must be positive** inorder for this equation to be **true** - $`\sqrt{x^2} = |x|`$ - You can subtract like terms only if they the radicals have the same **irrational factor**. Eg $`2 \sqrt{7} + 5 \sqrt{7} = 7 \sqrt{7}`$ ## Rationalizing Denominator - Its not proper to leave radicals in the denominator, so we can multiply the denominator by it self, inorder to get rid of the radical. - Eg $`\dfrac{\sqrt{7}}{\sqrt{3}} = \dfrac{\sqrt{7}}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} = \dfrac{\sqrt{21}}{3}`$ - 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. - Eg $`\dfrac{1+\sqrt{3}}{1-\sqrt{3}} \times \dfrac{1+\sqrt{3}}{1+\sqrt{3}} = \dfrac{(1+\sqrt{3})^2}{1-3} = \dfrac{1+2\sqrt{3}+3}{-2} = -4-\sqrt{3}`$ ## Introduction To Quadratic Equation - 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 $`D < 0`$, the quadratic equation has no real roots |Two distinct real roots|One real root|No real roots| |:---------------------:|:-----------:|:-----------:| |$`b^2-4ac>0`$|$`b^2-4ac=0`$|$`b^2-4ac<0`$| ## Complex Numbers - $`i = \sqrt{-1}`$. 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** ($`a \pm bi`$) ## Number Systems