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5.6 Quadratic Equations with Two Variables with Applications

4 min read•june 18, 2024

are a fundamental part of algebra, describing curves called parabolas. They pop up in everyday situations, from calculating profit margins to predicting . Understanding how to solve and graph these equations unlocks a world of practical problem-solving skills.

Mastering quadratics involves three key methods: graphing, , and using the . Each approach has its strengths, and knowing when to use which method is crucial. Real-world applications of quadratics include solving , working with consecutive integers, and tackling right triangle puzzles.

Quadratic Equations

Quadratic equation solving methods

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  • Graphing quadratic equations
    • Plot the equation y=ax2+bx+cy = ax^2 + bx + c on a coordinate plane by calculating points or using a graphing tool (, online graphing tool)
    • Locate the x-intercepts where the graph intersects the x-axis, representing the or solutions of the equation
  • Factoring quadratic equations
    • Rewrite the ax2+bx+c=0ax^2 + bx + c = 0 in
    • Decompose the quadratic expression into the product of two linear factors: (px+q)(rx+s)=0(px + q)(rx + s) = 0 (common factors: x2+2x+1=(x+1)(x+1)x^2 + 2x + 1 = (x + 1)(x + 1), x2−9=(x−3)(x+3)x^2 - 9 = (x - 3)(x + 3))
    • Utilize the : if the product of two factors equals zero, at least one factor must be zero
    • Determine the values of x by setting each factor equal to zero and solving the resulting linear equations
  • Quadratic formula
    • Apply the quadratic formula to solve equations in the form ax2+bx+c=0ax^2 + bx + c = 0 when factoring is not feasible or efficient
    • Quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients of the quadratic equation
    • Input the values of aa, bb, and cc into the formula and calculate the solutions by simplifying the expression (example: 2x2+7x−4=02x^2 + 7x - 4 = 0, a=2a = 2, b=7b = 7, c=−4c = -4)

Real-world applications of quadratics

  • Area problems
    • Define the area of a rectangle using the formula A=lwA = lw, where ll represents the length and ww represents the width
    • Formulate a quadratic equation by replacing the length and width with expressions based on the problem context (example: length is 3 units more than the width, l=w+3l = w + 3)
    • Determine the dimensions of the rectangle by solving the quadratic equation for the unknown variable
    • Assign the variable xx to represent the first integer in a sequence of consecutive integers (example: xx, x+1x + 1, x+2x + 2 for three consecutive integers)
    • Construct expressions for the subsequent integers in the sequence using xx as a reference
    • Develop a quadratic equation using the information provided in the problem and solve for xx to find the consecutive integers
    • Apply the : a2+b2=c2a^2 + b^2 = c^2, where cc represents the hypotenuse and aa and bb represent the legs of the right triangle
    • Replace the lengths of the sides with expressions based on the given information in the problem (example: one leg is 2 units shorter than the hypotenuse, a=c−2a = c - 2)
    • Generate a quadratic equation and solve for the missing variable to determine the lengths of the triangle's sides

Graphing Parabolas

Key features of parabola graphs

  • equation in standard form: y=ax2+bx+cy = ax^2 + bx + c
    • The value of aa determines the orientation and steepness of the parabola
      • For a>0a > 0, the parabola opens upward (positive quadratic term, )
      • For a<0a < 0, the parabola opens downward (negative quadratic term, inverted U-shaped graph)
    • This equation represents a , which is a type of
    • The vertex is the extreme point of the parabola, either a maximum or minimum point depending on the orientation
    • Vertex formula: (−b2a,f(−b2a))(\frac{-b}{2a}, f(\frac{-b}{2a})), where −b2a\frac{-b}{2a} is the x-coordinate and f(−b2a)f(\frac{-b}{2a}) is the y-coordinate of the vertex
    • The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two equal halves
    • Equation of the axis of symmetry: x=−b2ax = \frac{-b}{2a}, derived from the x-coordinate of the vertex
  • x-intercepts (roots or solutions)
    • The x-intercepts are the points where the parabola intersects the x-axis, representing the solutions to the quadratic equation
    • To determine the x-intercepts, set y=0y = 0 and solve the resulting quadratic equation using factoring or the quadratic formula (example: y=x2−4x−5y = x^2 - 4x - 5, xx-intercepts at x=5x = 5 and x=−1x = -1)

Properties of quadratic functions

  • and
    • The of a quadratic function is typically all real numbers, as the function is defined for any x-value
    • The range depends on the orientation of the parabola:
      • For a>0a > 0 (opens upward), the range is [yvertex,∞)[y_{vertex}, \infty)
      • For a<0a < 0 (opens downward), the range is (−∞,yvertex](-\infty, y_{vertex}]
    • The concavity of a parabola describes its curvature:
      • When a>0a > 0, the parabola is concave up (opens upward)
      • When a<0a < 0, the parabola is concave down (opens downward)
    • The concavity determines whether the vertex is a minimum or maximum point

Key Terms to Review (35)

Area problems: Area problems involve finding the size of a two-dimensional space enclosed within a given shape, often requiring the use of quadratic equations with two variables. These problems can include determining the area of rectangles, circles, triangles, and more complex shapes, and they frequently connect geometric concepts with algebraic expressions. Understanding area problems is crucial for solving real-world applications in fields like engineering, architecture, and design.
Axis of symmetry: The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. This concept is crucial when analyzing quadratic equations with two variables, as it helps to identify key properties such as the vertex and the direction of the parabola's opening. Understanding the axis of symmetry allows for a clearer interpretation of the graph and aids in solving real-world problems modeled by quadratic equations.
Binomial: A binomial is an algebraic expression containing exactly two distinct terms separated by a plus or minus sign. Each term can be a number, variable, or product of numbers and variables.
Completing the square: Completing the square is a mathematical technique used to transform a quadratic equation into a perfect square trinomial, making it easier to solve or analyze. This method allows for the derivation of the vertex form of a quadratic function and is essential for understanding the properties of parabolas, particularly their maximum or minimum values.
Concavity: Concavity refers to the direction a curve bends, indicating whether it opens upward or downward. This characteristic is essential for understanding the behavior of functions, especially when analyzing the shape of graphs related to quadratic equations with two variables. The concavity of a function helps determine local maxima and minima, which can have real-world applications in optimization problems.
Conic section: A conic section is a curve obtained by intersecting a cone with a plane. The nature of the intersection determines the type of conic section produced, which can be a circle, ellipse, parabola, or hyperbola. Each type of conic section has unique properties and equations that relate to quadratic equations with two variables, playing an essential role in various applications including physics, engineering, and astronomy.
Consecutive Integer Problems: Consecutive integer problems involve finding a sequence of integers that follow one after another without any gaps. These problems often require setting up equations based on relationships among the integers, typically expressed in terms of algebraic variables. They are commonly used in word problems and can be tied to various mathematical concepts, including properties of numbers and quadratic equations.
Directrix: The directrix is a fixed line used in the definition of a conic section, such as a parabola. It serves as a reference line that, along with a focus point, helps define the set of all points that form the curve. In the context of quadratic equations with two variables, the directrix is crucial for understanding the geometric properties of parabolas and their applications in various fields, including physics and engineering.
Discriminant: The discriminant is a mathematical expression that helps determine the nature of the roots of a quadratic equation. In the context of quadratic equations in two variables, it provides insights into whether the equation has real solutions, complex solutions, or repeated solutions. The discriminant is calculated using the formula $$D = b^2 - 4ac$$, where 'a', 'b', and 'c' are the coefficients of the quadratic equation in standard form.
Domain: The domain of a function is the set of all possible input values (typically represented as 'x') that the function can accept. It determines the range of values for which the function is defined and produces real numbers.
Domain: In mathematics, the domain refers to the complete set of possible values that a variable can take. It plays a crucial role in defining functions, inequalities, and equations as it establishes the input values for which these mathematical expressions are valid. Understanding the domain helps in determining where a function or equation can be applied and helps avoid undefined situations.
Factoring: Factoring is the process of breaking down an algebraic expression into a product of simpler factors. This concept is essential for simplifying expressions, solving equations, and understanding the relationships between variables, especially in polynomial equations where the roots can be identified through factoring.
Focus: In the context of quadratic equations, the focus refers to a specific point that plays a crucial role in defining the shape and properties of a conic section, particularly a parabola. It is located at a fixed distance from the vertex of the parabola along the axis of symmetry. The focus is essential in applications involving parabolic shapes, such as satellite dishes and the paths of projectiles, where the properties of quadratic equations help describe these real-world phenomena.
Graphing calculator: A graphing calculator is a handheld device capable of plotting graphs, solving equations, and performing complex calculations. These calculators can display multiple functions simultaneously and are equipped with features that allow users to manipulate data, making them essential tools for visualizing mathematical concepts and relationships.
Maximum/minimum point: A maximum or minimum point on a graph is a specific point where a function reaches its highest or lowest value, respectively. These points are crucial in understanding the behavior of quadratic equations with two variables, as they indicate the extreme values of the function, which can be relevant for optimization problems in various applications.
Monomial: A monomial is an algebraic expression consisting of a single term, which can be a constant, variable, or the product of constants and variables. It does not contain addition or subtraction operations.
Parabola: A parabola is a symmetric, U-shaped curve that is defined as the set of all points in a plane equidistant from a fixed point called the focus and a fixed line called the directrix. Parabolas can be represented mathematically by quadratic equations in two variables, typically in the form of $$y = ax^2 + bx + c$$ or $$x = ay^2 + by + c$$. This unique shape is important in various applications, such as projectile motion and reflective properties in physics.
Polynomial: A polynomial is an algebraic expression made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Common examples include linear equations and quadratic equations.
Polynomial: A polynomial is a mathematical expression that consists of variables, coefficients, and non-negative integer exponents. Polynomials can be represented in various forms, including as a sum of terms, where each term is a product of a coefficient and a variable raised to a power. They are foundational in algebra and play a significant role in understanding equations, particularly when solving for variables or analyzing relationships between different quantities.
Projectile motion: Projectile motion refers to the motion of an object that is thrown or projected into the air, subject to the influence of gravity and initial velocity. It is characterized by a curved path known as a parabola, which can be described mathematically using quadratic equations. Understanding projectile motion helps in analyzing various real-world scenarios, such as sports, engineering, and physics problems where objects follow a curved trajectory.
Pythagorean Theorem: The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.
Pythagorean theorem: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This relationship connects geometry and algebra, forming the basis for understanding distances and relationships within triangular structures and applications in real-world scenarios.
Quadratic equation: A quadratic equation is a polynomial equation of the second degree, typically expressed in the standard form as $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ represents the variable. These equations are characterized by their parabolic graphs, which can open upwards or downwards depending on the sign of the leading coefficient, $a$. Quadratic equations often appear in various applications, including physics, engineering, and economics, making them essential to understand for problem-solving in real-world scenarios.
Quadratic equations: Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. These equations can be graphed as parabolas and have applications in various fields.
Quadratic formula: The quadratic formula is a solution method for quadratic equations, given by the expression $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$, where $a$, $b$, and $c$ are coefficients from a standard form quadratic equation $ax^2 + bx + c = 0$. This formula provides the values of $x$ that satisfy the equation and is essential for finding solutions to equations that may not be easily factored.
Quadratic function: A quadratic function is a type of polynomial function that can be expressed in the standard form $$f(x) = ax^2 + bx + c$$, where $a$, $b$, and $c$ are constants and $a \neq 0$. This function forms a parabola when graphed, which can open either upward or downward depending on the sign of the coefficient $a$. Quadratic functions are essential in various applications, including physics, engineering, and finance, as they can model real-world phenomena like projectile motion and profit optimization.
Range: Range refers to the set of all possible output values (or dependent variable values) of a function, determined by the inputs in the domain. Understanding range is crucial as it helps to identify the limits of a function's output and how it behaves under different conditions, which can be connected to various mathematical concepts including inequalities, quadratic equations, and statistical measures.
Right triangle problems: Right triangle problems involve calculations and applications related to right triangles, which are triangles that contain a 90-degree angle. These problems often require using geometric principles and the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. These principles can be extended to quadratic equations when exploring relationships between variables in various contexts.
Roots: Roots refer to the values that satisfy an equation, particularly when it involves polynomial equations. They are crucial in understanding how functions behave and can indicate where a function crosses the x-axis. In various mathematical contexts, finding roots often involves operations, manipulations with exponents, and solving equations, highlighting their importance in diverse mathematical applications.
Standard form: Standard form refers to a way of writing numbers or equations in a specific format that makes them easy to understand and work with. In mathematical contexts, it often involves expressing numbers in a concise manner, such as writing large numbers in scientific notation or presenting quadratic equations in the form $$ax^2 + bx + c = 0$$ where a, b, and c are constants.
U-shaped graph: A U-shaped graph is a visual representation of a quadratic function that displays a parabolic curve opening upwards. This shape is significant because it illustrates the relationship between two variables in a quadratic equation, highlighting key features such as the vertex, axis of symmetry, and the behavior of the function as the input values change. Understanding U-shaped graphs is essential for analyzing and interpreting real-world situations modeled by quadratic equations.
Vertex: A vertex is a point where two or more curves, lines, or edges meet. In different contexts, it can represent a significant feature such as the peak of a parabola, a corner of a polygon, or a key point in graph theory. Understanding the concept of a vertex helps in analyzing the properties and relationships of various mathematical structures.
X-intercept: The x-intercept is the point on a graph where a function or relation crosses the x-axis, meaning that at this point, the value of y is zero. It is a critical concept in understanding linear equations, quadratic equations, and various types of functions, as it provides valuable information about their behavior and characteristics. The x-intercept can be found by setting the output (y-value) of an equation to zero and solving for the input (x-value).
Y-intercept: The y-intercept is the point where a graph intersects the y-axis, representing the value of the dependent variable when the independent variable is zero. This key feature helps to understand linear relationships, curves, and data trends, providing crucial information for graphing and analyzing equations across various mathematical contexts.
Zero Product Property: The Zero Product Property states that if the product of two or more factors equals zero, then at least one of the factors must be equal to zero. This principle is crucial for solving equations, particularly quadratic equations, as it allows for the determination of variable values when the equation is set to zero.
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5.7 Functions