highschool/Grade 10/Math/MPM2DZ/Unit 2: Quadratic Equations.md

Unit 2: Quadratic Equations

• $`\text{If } x \ge 0, x \in \mathbb{R}, \sqrt{x} \times \sqrt{x} = x`$

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

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 $`\dfrac{-b}{2a}`$

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 ($`a \pm bi`$)

Complex Number	Equivalent
$`i`$	$`\sqrt{-1}`$
$`i^2`$	$`-1`$
$`i^3`$	$`-\sqrt{-1}`$ or $`-i`$
$`i^4`$	$`1`$

Number Systems

• Natural Numbers $`\mathbb{N} = \{1,2,3, \cdots\}`$
• Whole Numbers $`\mathbb{W} = \{0, 1, 2, 3\cdots\}`$
• Integers $`(\mathbb{I}`$ or $`\mathbb{Z}) = \{\cdots, -2, -1, 0,1,2, \cdots\}`$
• Rational numbers $`(\mathbb{Q}) = \{\frac{a}{b}, a, b, \in \mathbb{I}, b =\not 0\}`$
• Irrational Numbers $`(\mathbb{Q} \prime)`$: any real number that cannot be written as $`\frac{a}{b}, a, b, \in \mathbb{I}, b =\not 0`$
• Real Numbers $`(\mathbb{R})`$: the set of $`\mathbb{Q} \cup \mathbb{Q} \prime`$
• Complex Numbers $`\mathbb{C}`$: any number that can be expressed in the form $`a+ib`$ (includes the set of real numbers)

Radical Equations

• Extraneous Sol $`\rightarrow`$ $`LS =\not RS`$
• Inadmissable Sol $`\rightarrow`$ Solutions you reject due to problem statement, eg negative length.
• Extraneous values occur because squaring both sides of an equation is not a reversible step.
• Make sure to check your work after working with radical equations, since squaring both sides is not a reversible step. Thus equations must be verified by pluging it back into the equation.
• Radical Equations are called that because the variable occurs under a radical sign. We rationalize the radical variable before continuing to slve the equation.