Solving Quadratics by Completing the Square

Professor Dave Explains
24 Sept 201707:32
EducationalLearning
32 Likes 10 Comments

TLDRIn this instructional video, Professor Dave introduces the mathematical technique known as 'completing the square,' a method crucial for solving polynomials that cannot be factored. By showcasing examples, the video demonstrates how adding specific terms can transform a polynomial into a perfect square, subsequently making it easier to solve by expressing it as the square of a binomial. Through step-by-step guidance, viewers learn how to manipulate equations to find solutions, even when dealing with coefficients or fractions. This video not only clarifies the process of completing the square but also emphasizes its practicality and applicability in solving complex polynomial equations.

Takeaways
  • 📚 Completing the square is a method used to solve polynomials that cannot be factored.
  • 📍 A polynomial can be considered a perfect square if it can be expressed as the square of a binomial.
  • 🌟 The process involves manipulating the equation to form a perfect square trinomial on one side.
  • 🔧 Adding a specific term to both sides of the equation allows us to transform it into a perfect square.
  • 🔮 For a polynomial x^2 + bx, the term added to complete the square is (\(\frac{b}{2}\)^2).
  • 💻 The coefficient of x is halved, squared, and added to both sides to facilitate completing the square.
  • 📱 Solving after completing the square involves taking the square root of both sides and then isolating x.
  • 📲 This technique yields two solutions, as taking the square root can result in both positive and negative roots.
  • 💾 When dealing with a coefficient on the x^2 term, divide the entire equation by that coefficient before completing the square.
  • 📸 The method provides a systematic approach to finding solutions for quadratics that are not easily factorable.
Q & A

  • What technique allows you to solve any polynomial equation?

    -Completing the square is a technique that will always allow you to solve a polynomial equation.

  • How do you complete the square?

    -To complete the square: 1) Bring all terms to one side, setting the equation equal to 0. 2) Take the coefficient of the x term, divide by 2, square it, and add that amount to both sides of the equation. This results in a perfect square trinomial that can be factored.

  • Why does completing the square work to solve polynomial equations?

    -Completing the square manufactures a perfect square trinomial, which can then be factored and solved using square root properties. This allows you to solve equations that may not factor directly.

  • If there is a coefficient on the x^2 term, what should you do before completing the square?

    -If there is a coefficient on the x^2 term, you should divide the entire equation by that coefficient first. This simplifies the remaining steps.

  • What do you do after completing the square?

    -After completing the square, you take the square root of both sides of the equation. Then, subtract the number you added when completing the square to solve for x.

  • Why do you get two answers when taking the square root?

    -When taking a square root, there are usually two solutions - a positive and negative one. This is because both the positive and negative root value will square to give you the original number under the radical.

  • Will solutions obtained from completing the square always be integers?

    -No, solutions obtained from completing the square may contain radicals or fractional values. They do not have to be neat integers.

  • What was the purpose of the first polynomial example (x^2 + 2x + 1)?

    -The first polynomial example demonstrates a perfect square trinomial that can be easily factored. This shows the logic behind deliberately creating a perfect square when solving equations by completing the square.

  • What do the factors created from completing the square represent?

    -The factors created (e.g. x + 1) represent the binomial that when squared gives the polynomial equation.

  • Why is this technique useful?

    -Completing the square allows you to solve any polynomial equation, even those that cannot be factored. This makes it an essential technique to learn.

Outlines
00:00
📚 Introduction to Completing the Square

This segment introduces the concept of completing the square, a technique used for solving polynomials that cannot be factored. Professor Dave begins with an example of a polynomial that is a perfect square (x^2 + 2x + 1) and demonstrates how it can be expressed as the square of a binomial (x + 1)^2. The process involves recognizing polynomials that are not perfect squares and strategically adding terms to create a perfect square, thus simplifying the solution of the equation. Two examples are provided to illustrate the method: one with a straightforward addition to complete the square and another involving a coefficient that must be halved and squared. The explanation emphasizes the utility of this technique in solving equations that cannot be factored by conventional means.

05:00
🔢 Advancing with Completing the Square

The second paragraph delves deeper into the process of completing the square, especially when dealing with coefficients and fractions. Professor Dave outlines a step-by-step method for dividing the entire equation by the coefficient of the x^2 term to simplify the equation before completing the square. This approach is exemplified with a polynomial that results in fractional values during the process. The method of finding the least common denominator to combine fractions and complete the square is explained, showcasing the technique’s adaptability to more complex equations. The segment reinforces the concept that the answers obtained may not always be neat but are mathematically accurate, demonstrating the versatility and effectiveness of completing the square in solving a wide range of polynomial equations.

Mindmap
Keywords
💡polynomial
A polynomial is an expression consisting of variables and coefficients. In the context of this video, polynomials refer to algebraic expressions with variables, often represented by x, and integer coefficients. Polynomials are discussed in relation to factoring and solving polynomial equations through techniques like completing the square.
💡factoring
Factoring refers to breaking down a polynomial into product of factors. The video contrasts factorable polynomials, which can be broken down into binomial factors, and non-factorable polynomials which require alternative solution techniques.
💡completing the square
Completing the square is a technique used to solve non-factorable polynomial equations. It involves adding a term to both sides of the equation to create a perfect square trinomial, allowing easy solution by taking square roots.
💡perfect square
A perfect square is a number that can be expressed as the square of another integer. The technique of completing the square manufactures perfect square polynomial expressions, hence its name. Examples of perfect squares are 4 (2 squared) and 49 (7 squared).
💡binomial
A binomial is a two-term polynomial expression such as x + 1. Completing the square generates binomial expressions through factorization of the resulting perfect square polynomial.
💡exponent
The exponent refers to the power that a quantity is raised to. Squaring is raising to an exponent of 2. The video discusses taking square roots, which involves an exponent of 1/2 to undo squaring.
💡coefficient
The coefficients are the numerical multipliers of the variable terms in a polynomial. To complete the square, the coefficient of the x term is divided by 2 and squared to give the extra term that must be added.
💡solution
The solutions refer to the values of x that satisfy the polynomial equation, making it equal to zero. Completing the square allows for the easy derivation of the solutions from the factorization.
💡expression
An expression in algebra describes a computation with numbers, variables, operations but no equality or inequality relation. Polynomials are algebraic expressions.
💡equation
An equation includes an equality/inequality relation between two expressions. The video focuses on solving polynomial equations through techniques like completing the square.
Highlights

Completing the square is a technique that will always work to solve polynomials.

If a polynomial can be factored into identical binomials, it must be a perfect square.

We can deliberately add a term to a polynomial to manufacture a perfect square, making it easy to solve.

To complete the square, take the X coefficient, cut it in half, square it, and add it to both sides of the equation.

When completing the square, the number you cut the X coefficient in half to becomes the number in the binomial.

If there is a coefficient on the X^2 term, divide everything by that coefficient before completing the square.

Don't worry if completing the square results in messy, unappealing solutions - they are still valid solutions.

A polynomial that can be factored into identical binomials must be a perfect square.

Adding a term to both sides of a polynomial equation can create a perfect square, making it easily solvable.

Cutting the X coefficient in half gives the term to put in the binomial when completing the square.

Always divide all terms by the X^2 coefficient before completing the square if it has a coefficient.

Don't expect integer or simplified radical solutions when completing the square.

A polynomial that factors into two identical binomials must form a perfect square.

Completing the square by adding a term manufactures a solvable perfect square polynomial.

The X coefficient divided by two gives the inner term when completing the square into a binomial.

Transcripts

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