Glossary

9.2 Solve Quadratic Equations by Completing the Square

3 min read•june 25, 2024

is a powerful technique for solving quadratic equations. It transforms a quadratic expression into a , making it easier to find solutions. This method is especially useful when dealing with equations that can't be easily factored.

By mastering completing the square, you'll gain a deeper understanding of quadratic relationships. It's not just about finding solutions – it also reveals the of quadratic functions, which is crucial for graphing and analyzing parabolas.

Completing the Square

Binomial to perfect square trinomial

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  • expression in the form ax2+bxax^2 + bx
    • aa represents of x2x^2 term
    • bb represents coefficient of xx term
  • Transform binomial into perfect square trinomial by adding square of half the coefficient of xx to both sides of equation
    • Calculate (b2)2(\frac{b}{2})^2 and add to both sides
    • Resulting trinomial on left side will be perfect square in the form (ax+b2)2(ax + \frac{b}{2})^2
    • Example: x2+6xx^2 + 6x becomes (x+3)2(x + 3)^2 by adding (62)2=32=9(\frac{6}{2})^2 = 3^2 = 9 to both sides

Completing the square with coefficient 1

  • in x2+bx+c=0x^2 + bx + c = 0, where a=1a = 1
  • Isolate variable terms on one side by subtracting cc from both sides
    • x2+bx=−cx^2 + bx = -c
  • Transform binomial x2+bxx^2 + bx into perfect square trinomial
    • Add (b2)2(\frac{b}{2})^2 to both sides of equation
    • Left side becomes (x+b2)2=−c+(b2)2(x + \frac{b}{2})^2 = -c + (\frac{b}{2})^2
  • Take square root of both sides
    • x+b2=±−c+(b2)2x + \frac{b}{2} = \pm \sqrt{-c + (\frac{b}{2})^2}
  • Solve for xx by subtracting b2\frac{b}{2} from both sides
    • x=−b2±−c+(b2)2x = -\frac{b}{2} \pm \sqrt{-c + (\frac{b}{2})^2}
    • Example: x2+6x+5=0x^2 + 6x + 5 = 0 becomes x=−3±−5+(62)2=−3±4=−3±2x = -3 \pm \sqrt{-5 + (\frac{6}{2})^2} = -3 \pm \sqrt{4} = -3 \pm 2, so x=−5x = -5 or x=−1x = -1

Completing the square for any coefficient

  • Quadratic equation in standard form ax2+bx+c=0ax^2 + bx + c = 0, where leading coefficient a≠1a \neq 1
  • Divide both sides by aa to make leading coefficient 1
    • x2+bax+ca=0x^2 + \frac{b}{a}x + \frac{c}{a} = 0
  • Isolate variable terms on one side by subtracting ca\frac{c}{a} from both sides
    • x2+bax=−cax^2 + \frac{b}{a}x = -\frac{c}{a}
  • Transform binomial x2+baxx^2 + \frac{b}{a}x into perfect square trinomial
    • Add (b2a)2(\frac{b}{2a})^2 to both sides of equation
    • Left side becomes (x+b2a)2=−ca+(b2a)2(x + \frac{b}{2a})^2 = -\frac{c}{a} + (\frac{b}{2a})^2
  • Take square root of both sides
    • x+b2a=±−ca+(b2a)2x + \frac{b}{2a} = \pm \sqrt{-\frac{c}{a} + (\frac{b}{2a})^2}
  • Solve for xx by subtracting b2a\frac{b}{2a} from both sides
    • x=−b2a±−ca+(b2a)2x = -\frac{b}{2a} \pm \sqrt{-\frac{c}{a} + (\frac{b}{2a})^2}
    • Example: 2x2+12x+7=02x^2 + 12x + 7 = 0 becomes x=−122(2)±−72+(122(2))2=−3±12=−3±22x = -\frac{12}{2(2)} \pm \sqrt{-\frac{7}{2} + (\frac{12}{2(2)})^2} = -3 \pm \sqrt{\frac{1}{2}} = -3 \pm \frac{\sqrt{2}}{2}

Other Methods for Solving Quadratic Equations

  • : Useful when the quadratic expression can be easily factored
  • : An alternative method derived from completing the square
  • : Helps determine the nature of roots (real or complex) without solving the equation
    • For ax2+bx+c=0ax^2 + bx + c = 0, the discriminant is b2−4acb^2 - 4ac
    • If discriminant > 0, there are two distinct
    • If discriminant = 0, there is one real root (repeated)
    • If discriminant < 0, there are two

Key Terms to Review (17)

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.
Binomial: A binomial is a polynomial expression that consists of two terms, typically connected by addition or subtraction operations. It is a fundamental concept in algebra that is essential for understanding and manipulating polynomial expressions.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the number of times a variable appears in a term or an 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.
Constant Term: The constant term is a numerical value that does not have a variable associated with it in a polynomial expression. It is the term that remains unchanged regardless of the value assigned to the variable(s) in the expression.
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.
General Form: The general form is a standardized way of expressing mathematical equations or expressions that allows for easy identification and manipulation of their underlying structure. This term is particularly relevant in the contexts of solving quadratic equations by completing the square and understanding the properties of ellipses.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It represents the scale or magnitude of the polynomial and plays a crucial role in various polynomial operations and properties.
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.
Perfect Square Trinomial: A perfect square trinomial is a special type of trinomial (an algebraic expression with three terms) that can be factored as the square of a binomial. This means that a perfect square trinomial can be expressed as the square of a sum or difference of two terms.
Quadratic Equation: A quadratic equation is a polynomial equation of the second degree, where the highest exponent of the variable is 2. It takes the general 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 algebra and have important applications in various fields, including physics, engineering, and economics.
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 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.
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 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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