🔟Elementary Algebra Unit 10 – Quadratic Equations

What Are Quadratic Equations?

• Quadratic equations are polynomial equations of degree 2, meaning the highest exponent of the variable is 2
• General form of a quadratic equation: $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$
• Quadratic equations can have up to two distinct solutions, also known as roots or zeros
• Solutions can be real numbers or complex numbers, depending on the values of $a$, $b$, and $c$
• Quadratic equations are used to model various real-world situations, such as projectile motion and optimization problems
• Solving quadratic equations involves finding the values of the variable that satisfy the equation
• Quadratic functions, represented by $f(x) = ax^2 + bx + c$, are closely related to quadratic equations

Key Components and Terms

• Coefficient: The numbers multiplied by the variable in a quadratic equation
  • $a$ is the leading coefficient, which multiplies $x^2$
  • $b$ is the coefficient of the linear term, which multiplies $x$
  • $c$ is the constant term
• Discriminant: The expression $b^2 - 4ac$, used to determine the nature and number of solutions
  • If the discriminant is positive, the equation has two distinct real solutions
  • If the discriminant is zero, the equation has one repeated real solution
  • If the discriminant is negative, the equation has two complex solutions
• Roots or zeros: The values of the variable that satisfy the quadratic equation, making it equal to zero
• Vertex: The point at which a quadratic function reaches its maximum or minimum value
  • The $x$-coordinate of the vertex is given by $-\frac{b}{2a}$
• Axis of symmetry: The vertical line that passes through the vertex of a quadratic function, dividing the graph into two symmetrical halves

Standard Form and How to Recognize It

• Standard form of a quadratic equation: $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$
• Recognizing standard form:
  • The equation must have an $x^2$ term, an $x$ term, and a constant term
  • The equation must be set equal to zero
  • The terms are typically arranged in descending order of degree (highest to lowest)
• Examples of equations in standard form: $2x^2 - 5x + 3 = 0$, $x^2 + 6x - 7 = 0$
• Equations not in standard form can be rearranged by isolating terms and setting the equation equal to zero
  • For example, $3x^2 = 6x - 9$ can be rewritten as $3x^2 - 6x + 9 = 0$
• Recognizing standard form is crucial for applying various solving methods and graphing techniques

Solving Methods: An Overview

• Factoring: Rewriting the quadratic expression as the product of two linear factors
  • Useful when the quadratic can be easily factored
  • Leads to finding the roots by setting each factor equal to zero
• Quadratic formula: A formula that directly calculates the roots of a quadratic equation
  • Useful for any quadratic equation, especially when factoring is difficult or impossible
  • The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Completing the square: Rewriting the quadratic equation in a form that allows for easier solving
  • Useful for deriving the quadratic formula and understanding the geometry of quadratic functions
• Graphing: Visualizing the quadratic function to estimate the roots and understand the behavior of the equation
  • Roots correspond to the $x$-intercepts of the graph
  • Vertex and axis of symmetry can be determined from the graph
• Choosing the appropriate solving method depends on the specific equation and the desired form of the solution

Factoring Quadratics

• Factoring is a method of solving quadratic equations by rewriting the expression as the product of two linear factors
• Goal is to find factors of the form $(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots of the equation
• Steps for factoring:
  1. Identify the coefficient of $x^2$ ($a$), the coefficient of $x$ ($b$), and the constant term ($c$)
  2. Find two numbers that multiply to give $ac$ and add to give $b$
  3. Rewrite the quadratic expression using these numbers
  4. Factor by grouping if necessary
• Examples of factoring:
  • $x^2 + 5x + 6 = (x + 2)(x + 3)$
  • $2x^2 - 7x - 15 = (2x + 3)(x - 5)$
• Special cases:
  • Perfect square trinomials: Quadratics of the form $a^2 \pm 2ab + b^2$, which can be factored as $(a \pm b)^2$
  • Difference of squares: Quadratics of the form $a^2 - b^2$, which can be factored as $(a + b)(a - b)$
• After factoring, set each linear factor equal to zero and solve for $x$ to find the roots

The Quadratic Formula

• The quadratic formula is a direct method for finding the roots of any quadratic equation
• Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation in standard form
• Discriminant ($b^2 - 4ac$) determines the nature of the roots:
  • If discriminant > 0, the equation has two distinct real roots
  • If discriminant = 0, the equation has one repeated real root
  • If discriminant < 0, the equation has two complex roots
• Steps for using the quadratic formula:
  1. Identify the values of $a$, $b$, and $c$ from the quadratic equation in standard form
  2. Substitute these values into the quadratic formula
  3. Simplify the expression under the square root (the discriminant)
  4. Calculate the two roots by adding and subtracting the square root term
• Examples of using the quadratic formula:
  • For $2x^2 - 5x + 2 = 0$, $a = 2$, $b = -5$, and $c = 2$. The roots are $x = \frac{5 \pm \sqrt{9}}{4}$, or $x = 1$ and $x = \frac{1}{2}$
• The quadratic formula is useful when factoring is difficult or when the roots are irrational or complex numbers

Graphing Quadratic Functions

• Quadratic functions are represented by the equation $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$
• The graph of a quadratic function is a parabola, which is symmetric about a vertical line called the axis of symmetry
• Key features of a quadratic graph:
  • Vertex: The point at which the parabola changes direction (maximum or minimum point)
    • $x$-coordinate of the vertex: $-\frac{b}{2a}$
    • $y$-coordinate of the vertex: $f(-\frac{b}{2a})$
  • Axis of symmetry: The vertical line that passes through the vertex, given by the equation $x = -\frac{b}{2a}$
  • $y$-intercept: The point at which the parabola intersects the $y$-axis, found by evaluating $f(0)$
  • $x$-intercepts (roots): The points at which the parabola intersects the $x$-axis, found by solving the quadratic equation $f(x) = 0$
• The sign of $a$ determines the direction of the parabola:
  • If $a > 0$, the parabola opens upward and has a minimum point
  • If $a < 0$, the parabola opens downward and has a maximum point
• Steps for graphing a quadratic function:
  1. Find the vertex by calculating $-\frac{b}{2a}$ and $f(-\frac{b}{2a})$
  2. Find the $y$-intercept by evaluating $f(0)$
  3. Find the $x$-intercepts by solving the quadratic equation $f(x) = 0$
  4. Plot the vertex, $y$-intercept, and $x$-intercepts (if any)
  5. Sketch the parabola using the plotted points and the axis of symmetry as a guide
• Graphing quadratic functions helps visualize the behavior of the equation and estimate the roots

Real-World Applications

• Projectile motion: Quadratic equations can model the path of objects thrown or launched, such as balls, rockets, or projectiles
  • The height of the object at time $t$ is given by $h(t) = -\frac{1}{2}gt^2 + v_0t + h_0$, where $g$ is the acceleration due to gravity, $v_0$ is the initial velocity, and $h_0$ is the initial height
• Area optimization: Quadratic equations can be used to find the dimensions of a rectangle or other shape that maximize or minimize the area given certain constraints
  • Example: Find the dimensions of a rectangular garden with a fixed perimeter that maximizes the area
• Business and economics: Quadratic functions can model profit, revenue, or cost as a function of the quantity of goods produced or sold
  • Example: A company's profit function is $P(x) = -2x^2 + 100x - 500$, where $x$ is the number of units produced. Find the production level that maximizes profit.
• Physics and engineering: Quadratic equations appear in problems involving energy, force, and motion
  • Example: The energy stored in a spring is given by $E = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement from the equilibrium position
• Quadratic equations and functions have numerous applications in various fields, making them essential tools for problem-solving and modeling real-world situations