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.