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2.5 Quadratic Equations

4 min read•june 24, 2024

Quadratic equations are a key part of algebra, describing relationships where one variable is squared. They pop up in many real-world situations, from physics to economics. Understanding how to solve them is crucial for tackling more complex math problems.

There are several ways to crack these equations, including , using the , and applying the . Each method has its strengths, and knowing when to use which can save you time and headaches in problem-solving.

Solving Quadratic Equations

Factoring techniques for quadratics

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  • Quadratic equations in ax2+bx+c=0ax^2 + bx + c = 0 where aa is not equal to 0 (a≠0a \neq 0)
  • states that if the product of two factors is zero, then at least one of the factors must be zero (ab=0ab = 0, then either a=0a = 0 or b=0b = 0, or both)
  • involves combining like terms and out common factors
    • Group terms with a common factor and factor out the (GCF) from each group
    • Factor out the GCF from the entire expression if possible (6x2+3x6x^2 + 3x can be factored as 3x(2x+1)3x(2x + 1))
  • Special factoring patterns include and perfect square trinomials
    • formula a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b) can be used to factor expressions like x2−9=(x+3)(x−3)x^2 - 9 = (x+3)(x-3)
    • Perfect square trinomials a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a+b)^2 and a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a-b)^2 factor into the square of a binomial (x2+6x+9=(x+3)2x^2 + 6x + 9 = (x+3)^2)
  • Solving quadratic equations by factoring involves factoring the expression and setting each factor equal to zero
    • Factor the quadratic expression completely
    • Set each factor equal to zero and solve the resulting linear equations to find the solutions (x2−5x+6=0x^2 - 5x + 6 = 0 factors to (x−2)(x−3)=0(x-2)(x-3) = 0, so x=2x = 2 or x=3x = 3)
    • These solutions are also known as the of the

Square root property in equations

  • states that if x2=ax^2 = a, then x=±ax = \pm \sqrt{a}
  • Isolating the squared term involves adding or subtracting terms to get the squared term alone on one side of the (x2+4=20x^2 + 4 = 20 becomes x2=16x^2 = 16)
  • Taking the square root of both sides of the equation and simplifying the result if possible (x2=16\sqrt{x^2} = \sqrt{16} becomes x=±4x = \pm 4)
  • Considering both positive and negative solutions is necessary because a squared term can have two square (9=±3\sqrt{9} = \pm 3)

Completing the square method

  • involves rewriting the in the form x2+bx=−cx^2 + bx = -c
    • Divide the of xx by 2 and square the result (b2)2(\frac{b}{2})^2
    • Add and subtract (b2)2(\frac{b}{2})^2 to the equation to create a
    • Factor the and isolate the squared term
    • Apply the square root property to solve for xx (x2+6x+5=0x^2 + 6x + 5 = 0 becomes (x+3)2=4(x+3)^2 = 4, so x=−3±4=−3±2x = -3 \pm \sqrt{4} = -3 \pm 2)
  • of a quadratic equation y=a(x−h)2+ky = a(x-h)^2 + k where (h,k)(h, k) is the and x=hx = h is the
  • quadratic equations involves identifying the vertex and , plotting additional points, and connecting them to form a
    • Use the equation or a table of values to find points on the graph
    • The will be symmetric about the axis of symmetry and open upward if a>0a > 0 or downward if a<0a < 0

Quadratic Formula and Applications

Quadratic formula applications

  • x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a} can be used to find solutions for any quadratic equation ax2+bx+c=0ax^2 + bx + c = 0
  • Δ=b2−4ac\Delta = b^2 - 4ac determines the nature of the solutions
    • If Δ>0\Delta > 0, the equation has two distinct real solutions (x2−5x+6=0x^2 - 5x + 6 = 0 has Δ=1\Delta = 1, so x=2x = 2 or x=3x = 3)
    • If Δ=0\Delta = 0, the equation has one repeated real solution (x2−6x+9=0x^2 - 6x + 9 = 0 has Δ=0\Delta = 0, so x=3x = 3)
    • If Δ<0\Delta < 0, the equation has no real solutions, only complex solutions (x2+2x+5=0x^2 + 2x + 5 = 0 has Δ=−16\Delta = -16, so no real solutions)
  • Simplifying the solutions by reducing square roots and fractions if possible
  • Applications of quadratic equations involve solving word problems related to , area, and other quadratic relationships
    • Identify the appropriate equation to model the situation, such as the height of a thrown ball h(t)=−16t2+64t+5h(t) = -16t^2 + 64t + 5
    • Interpret the solutions in the context of the problem, such as the time at which the ball reaches its maximum height or hits the ground

Additional Concepts in Quadratic Equations

  • A quadratic equation is a specific type of with a highest of 2
  • The coefficients in a quadratic equation are the numerical values that multiply the variables
  • Graphing quadratic equations results in a parabola, which is a U-shaped curve
  • The solutions to a quadratic equation can be found by solving the equation or by identifying where the graph of the function crosses the x-axis

Key Terms to Review (51)

Absolute value equation: An absolute value equation is an equation where the unknown variable appears inside absolute value bars, e.g., $|x| = a$. Solutions to these equations require considering both the positive and negative scenarios.
Absolute value function: An absolute value function is a type of piecewise function that returns the non-negative value of its input. It is denoted as $f(x) = |x|$ and has a V-shaped graph.
Axis of symmetry: The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. It passes through the vertex of the parabola and has the equation $x = -\frac{b}{2a}$ for a quadratic function in standard form.
Axis of Symmetry: The axis of symmetry is a line that divides a symmetrical figure, such as a parabola or absolute value function, into two equal halves. It represents the midpoint or line of reflection for the function, where the left and right sides are mirror images of each other.
Binomial coefficient: A binomial coefficient is a coefficient of any of the terms in the expansion of a binomial raised to a power, typically written as $\binom{n}{k}$ or $C(n,k)$. It represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order.
Co-vertex: The co-vertices of an ellipse are the endpoints of the minor axis. They are perpendicular to and lie at the midpoint of the major axis.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the magnitude or strength of the relationship between the variable and the overall expression. Coefficients are essential in various mathematical contexts, including polynomial factorization, linear equations, quadratic equations, and the graphing of polynomial functions.
Completing the square: Completing the square is a method to solve quadratic equations by converting them into a perfect square trinomial. This facilitates easier solving and helps in deriving the quadratic formula.
Completing the Square: Completing the square is a technique used to solve quadratic equations and transform quadratic functions into a more useful form. It involves rearranging a quadratic expression into a perfect square plus or minus a constant, allowing for easier analysis and manipulation of the equation or function.
Complex Roots: Complex roots are solutions to quadratic equations that have imaginary components. They occur when the discriminant of the equation is negative, indicating that the equation has no real solutions. Complex roots are expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ represents the imaginary unit, defined as the square root of -1.
Constant Term: The constant term is a numerical value in a polynomial or equation that does not depend on any variable. It is the term that remains unchanged regardless of the values assigned to the variables in the expression.
Difference of squares: The difference of squares is a specific type of polynomial that takes the form $a^2 - b^2$, which can be factored into $(a + b)(a - b)$. It is based on the property that the product of a sum and a difference of two terms results in the difference of their squares.
Difference of Squares: The difference of squares is a special type of polynomial expression where the terms are the difference between two perfect squares. This concept is particularly important in the context of factoring polynomials, working with rational expressions, solving quadratic equations, and understanding the properties of power functions and polynomial functions.
Discriminant: The discriminant is a value calculated from the coefficients of a quadratic equation $ax^2 + bx + c = 0$. It determines the nature and number of roots of the quadratic equation.
Discriminant: The discriminant is a value that determines the nature of the solutions to a quadratic equation. It provides information about the number and type of solutions, and is a crucial concept in the study of quadratic functions and the rotation of axes.
Equation: An equation is a mathematical statement that expresses the equality between two expressions or quantities. It represents a relationship between variables and constants, and is used to solve for unknown values or model real-world situations.
Exponent: An exponent indicates how many times a number, known as the base, is multiplied by itself. It is written as a small number to the upper right of the base.
Exponent: An exponent is a mathematical symbol that indicates the number of times a base number is multiplied by itself. It represents the power to which a number or variable is raised, and it is a fundamental concept in algebra, exponential functions, logarithmic functions, and other areas of mathematics.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions, often polynomials. It simplifies solving equations by expressing them as a product of factors.
Factoring: Factoring is the process of breaking down a polynomial or algebraic expression into a product of smaller, simpler expressions. It involves identifying common factors and using various techniques to express a polynomial as a product of its factors. Factoring is a fundamental algebraic skill that is essential for understanding and manipulating polynomials, rational expressions, quadratic equations, and other types of equations and functions.
Factoring by Grouping: Factoring by grouping is a technique used to factor polynomial expressions, particularly quadratic equations, by first grouping the terms in the polynomial and then identifying a common factor among the grouped terms. This method allows for the identification of the greatest common factor (GCF) within the polynomial, which can then be used to factor the expression into a more simplified form.
Function: A function is a mathematical relationship between two or more variables, where one variable (the dependent variable) depends on the value of the other variable(s) (the independent variable(s)). Functions are a fundamental concept in mathematics and are essential in understanding various topics in college algebra, including coordinate systems, quadratic equations, polynomial functions, and modeling using variation.
Graphing: Graphing is the visual representation of mathematical relationships, typically using a coordinate system to plot points, lines, curves, or other geometric shapes. It is a fundamental skill in mathematics that allows for the interpretation, analysis, and communication of quantitative information.
Greatest common factor: The greatest common factor (GCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. It is useful in simplifying fractions, factoring polynomials, and solving equations.
Greatest Common Factor: The greatest common factor (GCF) is the largest positive integer that divides two or more integers without a remainder. It is a fundamental concept in algebra that is essential for understanding real numbers, factoring polynomials, working with rational expressions, solving quadratic equations, and analyzing power and polynomial functions.
Leading coefficient: The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the end behavior of the polynomial function.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It is the first, or leading, coefficient in the standard form of a polynomial expression. The leading coefficient plays a crucial role in understanding and analyzing various topics in college algebra, including polynomials, quadratic equations, functions, and more.
Parabola: A parabola is a symmetric curve that represents the graph of a quadratic function. It can open upward or downward depending on the sign of the quadratic coefficient.
Parabola: A parabola is a curved, U-shaped line that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola, and has many important applications in mathematics, science, and engineering.
Perfect square trinomial: A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$.
Perfect Square Trinomial: A perfect square trinomial is a special type of polynomial expression that can be factored as the square of a binomial. It is a three-term polynomial in the form $a^2 + 2ab + b^2$, where $a$ and $b$ are real numbers.
Polynomial: A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents. It can be written in the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$ where $a_n, a_{n-1}, ..., a_1, a_0$ are constants and $n$ is a non-negative integer.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers. Polynomials are fundamental in various areas of mathematics, including algebra, calculus, and the study of functions.
Projectile Motion: Projectile motion is the motion of an object that is launched or projected into the air and moves under the influence of gravity, without the application of any additional force. It is a type of motion that follows a curved trajectory, typically a parabola, and is commonly observed in various contexts such as sports and everyday activities.
Quadratic equation: A quadratic equation is a second-degree polynomial equation in a single variable, typically written as $ax^2 + bx + c = 0$, where $a \neq 0$. The solutions to the quadratic equation are known as the roots of the equation.
Quadratic Equation: A quadratic equation is a polynomial equation of the second degree, where the highest exponent of the variable is 2. These equations take the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. Quadratic equations are fundamental in mathematics and have applications in various fields, including physics, engineering, and economics.
Quadratic formula: The quadratic formula is used to find the roots of a quadratic equation of the form $ax^2 + bx + c = 0$. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are polynomial equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. This formula provides a systematic way to find the solutions, or roots, of a quadratic equation.
Real Roots: Real roots refer to the solutions or values of a quadratic equation that are real numbers, as opposed to complex numbers. They represent the points where the graph of a quadratic function intersects the x-axis, providing the actual, tangible solutions to the equation.
Roots: Roots of a polynomial are the values of the variable that make the polynomial equal to zero. They are also known as solutions or zeros of the equation.
Roots: In mathematics, the term 'roots' refers to the solutions or values of a polynomial equation that make the equation equal to zero. Roots are an essential concept in various topics related to polynomial functions and equations, including quadratic equations, power functions, and the graphs of polynomial functions.
Square root property: The square root property is a method used to solve quadratic equations by taking the square root of both sides of the equation. It is particularly useful for equations in the form $ax^2 = c$.
Square Root Property: The square root property is a fundamental concept in solving quadratic equations. It states that the square root of a number is equal to both the positive and negative values that, when squared, result in the original number.
Standard form: Standard form of a linear equation in one variable is written as $Ax + B = 0$, where $A$ and $B$ are constants and $x$ is the variable. The coefficient $A$ should not be zero.
Standard Form: Standard form is a way of expressing mathematical equations or functions in a specific, organized format. It provides a consistent structure that allows for easier manipulation, comparison, and analysis of these mathematical representations across various topics in algebra and beyond.
Vertex: The vertex is a critical point in various mathematical functions and geometric shapes. It represents the point of maximum or minimum value, or the point where a curve changes direction. This term is particularly important in the context of quadratic equations, functions, absolute value functions, and conic sections such as the ellipse and parabola.
Vertex Form: The vertex form of a quadratic equation is a way to express the equation in a form that highlights the vertex of the parabolic graph. It is a useful representation that provides information about the maximum or minimum point of the parabola, as well as its orientation and symmetry.
Vertex form of a quadratic function: The vertex form of a quadratic function is given by $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola. It is useful for identifying the maximum or minimum point and the axis of symmetry.
Vieta's Formulas: Vieta's formulas are a set of algebraic identities that relate the coefficients of a quadratic equation to the roots of that equation. They provide a way to express the roots of a quadratic equation in terms of its coefficients, allowing for easier analysis and manipulation of quadratic functions.
Zero Product Property: The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. This property is fundamental in the context of factoring polynomials and solving quadratic equations.
Zero-product property: The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. It is commonly used to solve quadratic equations by factoring.
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2.6 Other Types of Equations