Solve by Completing the Square: Step-by-Step Technique

PreMath
30 Mar 201707:39
EducationalLearning
32 Likes 10 Comments

TLDRThis tutorial video demonstrates the process of solving a quadratic equation using the completing the square method. The instructor begins by isolating the variable x on one side and then divides the equation by the leading coefficient to simplify it. The next step involves taking half of the x coefficient, squaring it, and adding it to both sides to form a perfect square trinomial. The equation is then solved for x by taking the square root of both sides and isolating x, resulting in two solutions. The video concludes with the final solutions for x, clearly explained and presented.

Takeaways
  • 📚 The tutorial is about solving a quadratic equation using the completing the square method.
  • 🔍 The first step is to isolate the x variable by adding 15 to both sides of the equation.
  • ✂️ Eliminate the constant term by moving it to the other side of the equation.
  • 📉 Normalize the leading coefficient to 1 by dividing the entire equation by 3.
  • 🔢 Identify the coefficient of x, which is 4/3, and take half of it to prepare for completing the square.
  • 🧩 Multiply the half coefficient by itself to get the number to add to both sides of the equation.
  • 📈 Add the square of the half coefficient (2/3) to both sides to complete the square.
  • 🌐 Rewrite the left side of the equation as a perfect square trinomial and set it equal to the sum of 5 and 3/2.
  • 📝 Combine the fractions on the right side of the equation by finding a common denominator and adding them.
  • 🚀 Take the square root of both sides to solve for x, introducing a plus/minus sign.
  • 📌 Isolate x by subtracting 2/3 from both sides and simplify to find the two solutions for x.
  • 🎯 The final solutions for x are 5/3 and -3, which can be presented as a solution set: {-3, 5/3}.
Q & A

  • What method does the tutorial video demonstrate for solving a quadratic equation?

    -The tutorial video demonstrates solving a quadratic equation by completing the square method.

  • What is the first step in solving a quadratic equation by completing the square as shown in the video?

    -The first step is to ensure that only x variables are on the left-hand side, which involves adding 15 to both sides of the equation to eliminate the negative 15.

  • Why is it necessary to divide the equation by the leading coefficient in the video?

    -It is necessary to divide the equation by the leading coefficient to make it 1, which simplifies the equation and makes it easier to complete the square.

  • What does the video suggest doing with the coefficient of x after dividing by the leading coefficient?

    -The video suggests taking half of the coefficient of x (after dividing by the leading coefficient) and squaring it to complete the square.

  • How does the video simplify the fraction obtained after taking half of the coefficient of x?

    -The video simplifies the fraction by reducing it to its simplest form, which is 2/3, after initially obtaining 4/6.

  • What is the purpose of adding the square of the simplified fraction to both sides of the equation in the video?

    -The purpose is to create a perfect square trinomial on the left side of the equation, which allows for the equation to be expressed in the form of a squared binomial.

  • How does the video handle the constant term on the right side of the equation after adding the square of the fraction?

    -The video adds the constant term (5) to the square of the fraction (2/3 squared) and simplifies the sum to 49/9.

  • What mathematical operation is used to undo the square in the equation according to the video?

    -The video uses the operation of taking the square root of both sides of the equation to undo the square.

  • How does the video isolate x in the equation after taking the square root?

    -The video isolates x by subtracting 2/3 from both sides of the equation, which results in x = positive or negative (7/3 - 2/3).

  • What are the final solutions for x presented in the video?

    -The final solutions for x presented in the video are x = 5/3 and x = -3.

  • How does the video represent the solution set for the quadratic equation?

    -The video represents the solution set for the quadratic equation as -3, 5/3.

Outlines
00:00
📚 Introduction to Solving Quadratics by Completing the Square

The video tutorial begins by introducing the process of solving a quadratic equation using the completing the square method. The presenter emphasizes the importance of having x terms on one side and constants on the other, which involves adding 15 to both sides to eliminate the negative 15. The equation is then simplified by dividing by 3 to make the leading coefficient 1, resulting in x² + (4/3)x = 5. The presenter instructs viewers to take half of the x coefficient, which is (2/3), square it to get (4/9), and add this to both sides of the equation. This step transforms the left side into a perfect square trinomial, allowing the equation to be rewritten as (x + 2/3)² = 5 + 4/9. The presenter then demonstrates how to simplify the right side of the equation by adding the fractions, resulting in (x + 2/3)² = 49/9.

05:02
🔍 Final Steps to Solve the Quadratic Equation

The tutorial continues with the final steps to solve the quadratic equation. The presenter explains how to take the square root of both sides to eliminate the square, introducing a plus or minus sign, leading to x + 2/3 = ±√(49/9). The square roots of 49 and 9 are simplified to 7 and 3, respectively, resulting in x + 2/3 = ±7/3. To isolate x, the presenter subtracts 2/3 from both sides, yielding x = -7/3 ± 2/3. The solutions are then split into two cases: x = 5/3 (positive case) and x = -3 (negative case). The presenter concludes by stating the solution set as x = -3, 5/3, and thanks the viewers for watching.

Mindmap
Keywords
💡Quadratic Equation
A quadratic equation is a polynomial equation of the second degree, typically in the form of ax² + bx + c = 0, where x represents an unknown variable, and a, b, and c are constants with the condition that a ≠ 0. In the context of the video, the quadratic equation is the central theme, as the tutorial focuses on solving such an equation using a specific method.
💡Completing the Square
Completing the square is a method used to solve quadratic equations. It involves manipulating the quadratic and linear terms on one side of the equation to form a perfect square trinomial, which can then be solved by taking the square root of both sides. In the video, this method is used to solve the given quadratic equation, making it a key concept in the tutorial.
💡Leading Coefficient
The leading coefficient in a quadratic equation is the coefficient of the x² term. It is crucial in the process of completing the square because, for the method to work effectively, the leading coefficient should be set to 1. In the script, the leading coefficient is initially 3, and the equation is divided by 3 to make the leading coefficient 1.
💡Variable
In mathematics, a variable represents an unknown quantity that can take on different values. In the context of the video, 'x' is the variable in the quadratic equation that the tutorial aims to solve for. The script ensures that all 'x' terms are on one side of the equation to facilitate the solving process.
💡Coefficient
A coefficient is a numerical factor that multiplies a variable in an algebraic expression. In the script, the coefficient for 'x' is initially 4/3, and the process involves taking half of this coefficient to complete the square, which is a crucial step in the method demonstrated.
💡Isolating the Variable
Isolating the variable means getting the variable alone on one side of the equation, which is a common goal in solving equations. In the video, after completing the square, the script involves isolating 'x' by performing operations such as subtracting 2/3 from both sides of the equation.
💡Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number. In the script, taking the square root is a step used to undo the square that was created when completing the square, allowing the solver to find the value of 'x'.
💡Plus/Minus Sign
The plus/minus sign (±) indicates that there are two possible solutions to an equation when taking the square root of both sides. In the video script, after taking the square root of the completed square, a plus/minus sign is introduced to account for both the positive and negative roots of the equation.
💡Common Denominator
A common denominator is a denominator that is the same for two or more fractions, allowing them to be added or subtracted. In the script, when simplifying the solutions for 'x', the common denominator of 3 is used to combine terms and find the final solutions.
💡Solution Set
The solution set refers to the set of all possible solutions to an equation. In the context of the video, after finding the values for 'x', the script presents the solution set as {-3, 5/3}, indicating the two values of 'x' that satisfy the original quadratic equation.
Highlights

Introduction to solving quadratic equations using the completing the square method.

Ensuring only x variables are on the left side by adding 15 to both sides of the equation.

Dividing the equation by the leading coefficient of 3 to simplify the equation.

Focusing on the coefficient of x (4/3) and taking half of it to prepare for completing the square.

Multiplying the half coefficient by 1/2 to get 2/3 and squaring it to add to both sides of the equation.

Transforming the left side of the equation into a complete square form.

Adding 5 and 4/9 on the right side of the equation to simplify the expression.

Cross-multiplying to add fractions and simplify the equation.

Taking the square root of both sides to undo the square and isolate x.

Applying the plus/minus sign to both sides after taking the square root.

Isolating x by subtracting 2/3 from both sides.

Solving for x by simplifying the fractions and finding the common denominator.

Finding the first solution for x as 5/3.

Calculating the second solution for x as -3.

Presenting the final solution set as x = -3, 5/3.

Conclusion and thanks for watching the tutorial.

Transcripts

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