Glossary

9.3 Solve Quadratic Equations Using the Quadratic Formula

3 min read•june 25, 2024

Quadratic equations are fundamental concept in algebra. They represent relationships where one variable is squared, leading to curved graphs called parabolas. Understanding how to solve these equations is crucial for tackling more complex math problems.

The is a powerful tool for solving quadratic equations. It works for all quadratics, even when other methods fail. By analyzing the , we can determine the number and nature of solutions, giving us insights into the equation's behavior.

Solving Quadratic Equations

Application of Quadratic Formula

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  • Quadratic Formula x=−[b](https://www.fiveableKeyTerm:b)±b2−4ac2ax = \frac{-[b](https://www.fiveableKeyTerm:b) \pm \sqrt{b^2 - 4ac}}{2a} solves quadratic equations in ax2+bx+[c](https://www.fiveableKeyTerm:c)=0ax^2 + bx + [c](https://www.fiveableKeyTerm:c) = 0
  • Apply Quadratic Formula by identifying coefficients aa, bb, and cc, substituting into formula, simplifying discriminant b2−4acb^2 - 4ac, calculating two solutions by adding and subtracting square root term
  • ±\pm symbol represents two possible solutions: one with addition (+)(+) and one with subtraction (−)(-)
  • Example: Solve 2x2−5x−3=02x^2 - 5x - 3 = 0 using Quadratic Formula
    • a=2a = 2, b=−5b = -5, c=−3c = -3
    • x=−(−5)±(−5)2−4(2)(−3)2(2)x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)}
    • x=5±25+244=5±494=5±74x = \frac{5 \pm \sqrt{25 + 24}}{4} = \frac{5 \pm \sqrt{49}}{4} = \frac{5 \pm 7}{4}
    • Two solutions: x=5+74=3x = \frac{5 + 7}{4} = 3 and x=5−74=−12x = \frac{5 - 7}{4} = -\frac{1}{2}
  • These solutions are also called the of the quadratic equation

Interpretation of discriminant

  • Discriminant b2−4acb^2 - 4ac determines nature and number of solutions for quadratic equation
    • Positive discriminant (b2−4ac>0)(b^2 - 4ac > 0) indicates two distinct real solutions ( intersects x-axis at two points)
    • Zero discriminant (b2−4ac=0)(b^2 - 4ac = 0) indicates one repeated real solution (parabola touches x-axis at one point)
    • Negative discriminant (b2−4ac<0)(b^2 - 4ac < 0) indicates two complex solutions involving imaginary unit ii where i2=−1i^2 = -1 (parabola does not intersect x-axis)
  • Example: Determine number and nature of solutions for 3x2+6x+2=03x^2 + 6x + 2 = 0
    • a=3a = 3, b=6b = 6, c=2c = 2
    • Discriminant =62−4(3)(2)=36−24=12>0= 6^2 - 4(3)(2) = 36 - 24 = 12 > 0
    • Two distinct real solutions exist

Choosing the Best Method

Methods for solving quadratics

  • most efficient when quadratic has integer coefficients, leading coefficient aa is ±1\pm 1, constant term cc factors into two integers with sum bb
    • Example: x2+7x+12=0x^2 + 7x + 12 = 0 factors as (x+3)(x+4)=0(x + 3)(x + 4) = 0
  • useful for rewriting equation in to identify and coordinates, deriving Quadratic Formula
    • Example: x2+6x+5=0x^2 + 6x + 5 = 0 becomes (x+3)2−4=0(x + 3)^2 - 4 = 0 in vertex form
  • Quadratic Formula most general method, applies when equation cannot be factored, coefficients are fractions or decimals, exact solution needed
    • Example: 2x2−3x−12=02x^2 - \sqrt{3}x - \frac{1}{2} = 0 solved using Quadratic Formula

Graphical Representation and Analysis

  • Quadratic equations represent parabolas when graphed in the coordinate plane
  • The solutions (roots) of a quadratic equation correspond to the x-intercepts of its graph
  • a quadratic function can provide visual insight into the nature of its solutions
  • Quadratic equations are a specific type of equation of degree 2

Key Terms to Review (19)

A: The variable 'a' is a fundamental component in the study of quadratic equations and ellipses. It represents a constant value that is used in the mathematical expressions and formulas related to these topics.
Axis of Symmetry: The axis of symmetry is a line that divides a symmetric figure, such as a parabola, into two equal halves. It represents the midpoint or center of the symmetric figure, where the function changes direction from increasing to decreasing or vice versa.
B: The variable 'b' is a commonly used parameter that represents a constant value or coefficient in various mathematical contexts. It is a fundamental component in equations and formulas, playing a crucial role in the analysis and understanding of linear, quadratic, and elliptical relationships.
C: The variable 'c' is a commonly used letter to represent a constant or a coefficient in various mathematical contexts, including the solving of quadratic equations using the quadratic formula and the equation of an ellipse. It serves as a placeholder for a specific numerical value that remains fixed throughout the problem or equation.
Completing the Square: Completing the square is a technique used to solve quadratic equations by transforming them into a perfect square form. This method involves adding a constant to both sides of the equation to create a perfect square on one side, allowing for easier factorization and solution of the equation.
Complex Roots: Complex roots refer to the solutions of a quadratic equation that are complex numbers, meaning they have both a real and an imaginary component. These roots arise when the discriminant of the quadratic equation is negative, indicating that the equation has no real solutions.
Discriminant: The discriminant is a mathematical expression that determines the nature of the solutions to a quadratic equation. It plays a crucial role in understanding the behavior and characteristics of polynomial equations, quadratic equations, and their graphical representations.
Factoring: Factoring is the process of breaking down a polynomial expression into a product of simpler polynomial expressions. This technique is widely used in various areas of mathematics, including solving equations, simplifying rational expressions, and working with quadratic functions.
Graphing: Graphing is the visual representation of mathematical relationships, often using a coordinate plane or graph paper, to depict the behavior and characteristics of functions, equations, and data. It is a fundamental tool in mathematics and various scientific disciplines for analyzing, interpreting, and communicating quantitative information.
Imaginary Roots: Imaginary roots are the solutions to a quadratic equation that are complex numbers, meaning they have a real and an imaginary component. These roots arise when the discriminant of the quadratic equation is negative, indicating that the equation has no real solutions.
Parabola: A parabola is a curved, U-shaped line or surface that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola. Parabolas have many important applications in mathematics, physics, and engineering.
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 algebra and play a crucial role in various mathematical topics covered in this course.
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.
Radicand: The radicand is the quantity or expression under the radical sign in a radical expression. It represents the value or number that is to be operated on by the radical symbol, such as the square root or cube root.
Real Roots: Real roots are the solutions to an equation that are real numbers, as opposed to complex numbers. They represent the points where a graph of the equation intersects the x-axis. Real roots are an important concept in understanding the behavior and properties of quadratic equations and functions.
Roots: Roots refer to the values of a variable that satisfy an equation or inequality. They represent the solutions to polynomial expressions, where the roots are the x-values that make the equation or inequality equal to zero. Roots are a fundamental concept in algebra, as they are essential for understanding and solving various types of polynomial functions and equations.
Standard Form: The standard form of an equation is a specific way of writing the equation that provides a clear and organized structure, making it easier to analyze and work with the equation. This term is particularly relevant in the context of linear equations, quadratic equations, and other polynomial functions.
Vertex: The vertex of a function or graph is the point where the graph changes direction, either from decreasing to increasing or from increasing to decreasing. It is the turning point of the graph and represents the maximum or minimum value of the function.
Vertex Form: The vertex form of a quadratic equation is a way of expressing the equation in a specific format that highlights the vertex of the parabolic graph. The vertex form emphasizes the coordinates of the vertex, which are the point where the parabolic curve changes direction from increasing to decreasing or vice versa.
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