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Essential Concepts

Quadratic Equations

• A quadratic equation is a type of polynomial equation of the second degree, which means it involves at least one term that is squared (i.e., raised to the power of two).
• Many quadratic equations can be solved by factoring when the equation has a leading coefficient of [latex]1[/latex] or if the equation is a difference of squares. The zero-factor property is then used to find solutions.
• Many quadratic equations with a leading coefficient other than [latex]1[/latex] can be solved by factoring using the grouping method.
• Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution.
• Completing the square is a method of solving quadratic equations when the equation cannot be factored.
• A highly dependable method for solving quadratic equations is the quadratic formula based on the coefficients and the constant term in the equation.
• The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation.

Other Types of Equations

• A rational equation contains at least one rational expression (a fraction where the numerator, denominator, or both are polynomials).
• Eliminate all denominators by multiplying both sides by the least common denominator (LCD).
• Least Common Denominator (LCD) is an expression that contains the highest power of all factors in the denominators, used to clear fractions by canceling out terms in the denominator.
• When a rational equation is set up as a proportion, use cross-multiplication to solve it without finding the LCD.
• Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve a radical equation, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1.
• Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping.
• We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index.
• To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value.
• Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form.
• Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value with the inequality symbol flipped.
  • [latex]|X|> k[/latex] which is equivalent to: [latex]X< -k\text{, or }X> k[/latex]
• Absolute value inequality solutions can be verified by graphing. We can check the algebraic solutions by graphing as we cannot depend on a visual for a precise solution.

Applications of Non-Linear Equations

• Non-linear equations model complex real-world relationships
  • More accurate for natural phenomena
  • Used in physics, engineering, economics, environmental science
• Projectile motion follows a parabolic trajectory

Key Equations

Standard Form of Quadratic Equation	[latex]ax^2 + bx + c = 0[/latex], where [latex]a \neq 0[/latex]
Zero Product Property	[latex]\text{If }a \cdot b = 0, \text{ then }a = 0\text{ or }b = 0[/latex]
Square Root Property	[latex]\text{If }x^2 = k\text{, then }x = \pm\sqrt{k}[/latex]
Perfect Square Trinomials	[latex]a^2 + 2ab + b^2 = (a + b)^2[/latex] [latex]a^2 - 2ab + b^2 = (a - b)^2[/latex]
Quadratic Formula	[latex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/latex]
Pythagorean Theorem	[latex]a^2 + b^2 = c^2[/latex]
Rational Exponents	[latex]a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m[/latex]
Polynomial Standard Form	[latex]a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0[/latex]
Cross Multiplication (Rationals)	[latex]\text{If }\frac{a}{b} = \frac{c}{d}\text{, then }ad = bc[/latex]
Absolute Value Definition	[latex]|x| = \begin{cases} x & \text{if }x \geq 0 \\ -x & \text{if }x < 0 \end{cases}[/latex]
Absolute Value Inequalities (Less Than)	[latex]|X| < k \text{ is equivalent to } -k < X < k[/latex]
Absolute Value Inequalities (Greater Than)	[latex]|X| > k \text{ is equivalent to } X < -k \text{ or } X > k[/latex]
Zero Product Property	[latex]\text{If }a \cdot b = 0\text{, then }a = 0\text{ or }b = 0[/latex]
Projectile Motion Height	[latex]h = at^2 + bt + c[/latex]
Rectangle Area	[latex]A = lw[/latex]
Square Area	[latex]A = s^2[/latex]
Triangle Area	[latex]A = \frac{1}{2}bh[/latex]
Circle Area	[latex]A = \pi r^2[/latex]
Rectangle Perimeter	[latex]P = 2L + 2W[/latex]
Box Volume	[latex]V = LWH[/latex]
Sphere Volume	[latex]V = \frac{4}{3}\pi r^3[/latex]
Pendulum Period	[latex]T = 2\pi\sqrt{\frac{L}{g}}[/latex]

Glossary

absolute value equation

an equation in which the variable appears in absolute value bars, typically with two solutions, one accounting for the positive expression and one for the negative expression

area

amount of space inside a two-dimensional shape

binomial
An expression with two terms, such as \( x + 1 \), which should be treated as an individual factor when finding the LCD.

completing the square

a process for solving quadratic equations in which terms are added to or subtracted from both sides of the equation in order to make one side a perfect square

cross-multiplication

A method for solving rational equations set up as a proportion by multiplying terms across the equal sign.

excluded values

Values that make the denominator in a rational expression equal to zero, which must be excluded from the solution set.

Least Common Denominator (LCD)

An expression that contains the highest power of all factors in the denominators, used to clear fractions by canceling out terms in the denominator.

perimeter

distance around the outside of a shaped

polynomial equation
an equation containing a string of terms including numerical coefficients and variables raised to whole-number exponents

projectile motion

motion of an object thrown or launched near Earth’s surface

Pythagorean Theorem

a theorem that states the relationship among the lengths of the sides of a right triangle, used to solve right triangle problems

quadratic equation

an equation containing a second-degree polynomial; can be solved using multiple methods

quadratic formula

a formula that will solve all quadratic equations

radical equation

an equation containing at least one radical term where the variable is part of the radicand

rate

speed or frequency at which something occurs

rational equation

An equation that contains at least one rational expression, where the variable appears in at least one of the denominators.

rational expression

The ratio or quotient of two polynomials, e.g., \( \frac{x+1}{x^2-4} \), \( \frac{1}{x-3} \), or \( \frac{4}{x^2+x-2} \).

rational number

The ratio of two numbers, such as \( \frac{3}{4} \) or \( \frac{7}{2} \).

square root property

one of the methods used to solve a quadratic equation in which the [latex]{x}^{2}[/latex] term is isolated so that the square root of both sides of the equation can be taken to solve for x

volume

amount of space inside a three-dimensional object

zero-product property

the property that formally states that multiplication by zero is zero so that each factor of a quadratic equation can be set equal to zero to solve equations