How to Factor any Quadratic Equation Easily - Trick for factorising

tecmath
29 Mar 201806:43
EducationalLearning
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TLDRIn this Tech Math Channel video, the presenter teaches viewers how to easily factorize quadratic equations. By identifying coefficients a, b, and c, and finding two numbers that multiply to 'ac' and add to 'b', the equation can be factored into simpler forms. The method is demonstrated with various examples, showing how to flip signs and divide by the leading coefficient to find the solutions for x. The video is designed to be informative and engaging, with a step-by-step approach to solving quadratic equations.

Takeaways
  • 📚 The video teaches a simple method to factorize quadratic equations.
  • 🔍 Identify the coefficients 'a', 'b', and 'c' from the quadratic equation in the form of ax² + bx + c = 0.
  • 🤔 Multiply 'a' and 'c' to find a product that will help in finding two factors.
  • 🔢 Look for two numbers that multiply to the product of 'a' and 'c' and add up to 'b'.
  • 🔄 Flip the signs of the identified numbers to get the correct factors for the equation.
  • ✂️ Divide each factor by the coefficient 'a' to find the possible values for 'x'.
  • 📉 Example given: For 2x² + 7x - 4 = 0, the factors are -8 and +1, leading to solutions x = -4 and x = 1/2.
  • 📈 Another example: For 2x² - 2x - 12 = 0, the factors are -4 and +6, resulting in solutions x = -2 and x = 3.
  • 😮 Tricky case: For -10x² + 13x + 3 = 0, the correct factors are 15 and -2, not 10 and 3, leading to solutions x = 1 and x = 1/2.
  • 📌 Final example: For 6x² - x - 2 = 0, the factors are -4 and +3, giving solutions x = 2/3 and x = -2.
  • 👍 The method is a straightforward approach to solving quadratic equations by factorization.
Q & A

  • What is the first step in factorizing the quadratic equation 2x^2 + 7x - 4 = 0?

    -The first step is to identify and label the coefficients: a = 2, b = 7, and c = -4.

  • How do you determine the product of the coefficients in the equation 2x^2 + 7x - 4 = 0?

    -You multiply the first coefficient (a) by the constant term (c). In this case, 2 * -4 = -8.

  • What is the significance of the middle coefficient (b) in the factorization process?

    -The middle coefficient (b) is used to find two numbers that multiply to the product of a and c (-8 in this case) and add up to b (7 in this case).

  • Which two numbers satisfy the conditions for the equation 2x^2 + 7x - 4 = 0?

    -The numbers 8 and -1 satisfy the conditions because 8 * -1 = -8 and 8 + (-1) = 7.

  • How do you adjust the signs of the factors during the process?

    -You flip the signs of the factors: positive becomes negative and negative becomes positive.

  • What is the final step after adjusting the signs of the factors?

    -You divide the adjusted factors by the coefficient a. For example, the factors become -8/2 and 1/2.

  • How do you solve the quadratic equation 2x^2 - 2x - 12 = 0?

    -You follow the same steps: identify a, b, and c (a = 2, b = -2, c = -12), find two numbers that multiply to a*c (-24) and add to b (-2), which are 4 and -6, flip the signs, and divide by a.

  • What is the trick to solving the equation -10x^2 + 13x + 3 = 0?

    -Be cautious with the factors; the correct factors that multiply to -30 and add to 13 are 15 and -2.

  • How do you simplify the factors in the equation -10x^2 + 13x + 3 = 0?

    -You flip the signs of the factors (15 becomes -15, -2 becomes 2) and then divide by the coefficient a (-10).

  • What are the solutions for the equation 6x^2 - x - 2 = 0?

    -Identify a, b, and c (a = 6, b = -1, c = -2), find the numbers that multiply to -12 and add to -1, which are -4 and 3, flip the signs, and divide by 6 to get the solutions x = 2/3 and x = -1/3.

Outlines
00:00
📚 Introduction to Factorizing Quadratic Equations

In this segment, the Tech Math Channel introduces viewers to a simple method for factorizing quadratic equations. The presenter uses the example of 2x^2 + 7x - 4 = 0 to demonstrate the process. The key is to identify coefficients a, b, and c, then find two numbers that multiply to give the product of a and c and add up to b. For the given example, the numbers are -1 and 8, which after flipping signs, lead to the solutions x = -4 or x = 0.5.

05:04
🔍 Factorizing Quadratics with More Complex Coefficients

This paragraph delves into factorizing more complex quadratic equations, starting with 2x^2 - 2x - 12 = 0. The method involves multiplying a and c to get -24 and finding factors of -24 that sum to b, which is -2. The factors identified are 4 and -6, which after sign flipping give the solutions x = -2 or x = 3. The presenter then tackles an even more complex equation, -10x^2 + 13x + 3 = 0, cautioning viewers about the trickier factorization process. The correct factors of -30 that sum to 13 are 15 and -2, leading to the solutions x = 1.5 or x = 2/10.

🎯 Solving a Quadratic with No Middle Term

The final paragraph of the script addresses the factorization of a quadratic equation without a middle term: 6x^2 - x - 2 = 0. The presenter applies the same method, multiplying a and c to get -12 and identifying factors of -12 that sum to b, which is -1. The factors are -4 and 3, which after sign flipping result in the solutions x = 2/3 or x = -2. The presenter wraps up the video by summarizing the process and inviting feedback from viewers.

Mindmap
Keywords
💡Factorize
Factorize refers to the process of breaking down a complex expression or equation into a product of simpler, more fundamental expressions. In the context of the video, factorization is used to solve quadratic equations by finding two binomials that, when multiplied, give the original equation. For example, the script discusses factorizing the equation 2x^2 + 7x - 4 = 0 into (2x - 1)(x + 4).
💡Quadratic Equations
A quadratic equation is a polynomial equation of degree two, typically in the form ax^2 + bx + c = 0, where a, b, and c are constants. The video's main theme revolves around simplifying the process of solving these equations by factorization. The script provides examples of various quadratic equations and demonstrates the step-by-step process of finding their solutions.
💡Coefficients
In the context of the script, coefficients refer to the numerical factors that multiply the variables in a polynomial equation. For a quadratic equation, 'a', 'b', and 'c' are the coefficients of x^2, x, and the constant term, respectively. The script uses the coefficients to guide the factorization process, as seen when it multiplies 'a' and 'c' to find a product that will help in finding the factors.
💡Middle Coefficient
The middle coefficient in a quadratic equation is the coefficient of the linear term (x), which is represented by 'b' in the equation ax^2 + bx + c. The script mentions using the middle coefficient to find two numbers that, when multiplied, give the product of 'a' and 'c', and when added, give 'b'. This is a crucial step in the factorization method demonstrated.
💡Factors
Factors are numbers or expressions that can be multiplied together to produce another number or expression. In the script, the host is looking for two factors of the product of 'a' and 'c' that also add up to the middle coefficient 'b'. These factors are essential for rewriting the quadratic equation in its factorized form, as shown in the examples provided.
💡Solving Quadratic Equations
Solving quadratic equations involves finding the values of the variable that make the equation true. The video script demonstrates a method of solving quadratic equations by factorization, which involves rewriting the equation in a form that can be easily decomposed into binomials, whose solutions are then found by setting each binomial equal to zero.
💡Binomials
A binomial is an algebraic expression with two terms, typically in the form (ax + b). In the context of the script, the quadratic equation is factorized into a product of two binomials. The process involves finding two numbers that meet specific multiplication and addition criteria, which then form the binomials that solve the equation.
💡Product
In mathematics, a product is the result of multiplying two or more numbers or expressions. The script refers to the product of the coefficients 'a' and 'c' to establish a target for the factors that need to be found. The product is a key part of the factorization process, as it helps determine the correct pair of factors.
💡Addition
Addition is the process of combining two or more numbers to find their total or sum. In the script, the sum of the two factors found is required to equal the middle coefficient 'b'. This is part of the criteria used to identify the correct factors for the quadratic equation's factorization.
💡Multiplication
Multiplication is the mathematical operation of scaling one number by another. In the script, the host multiplies the factors to ensure they produce the product of 'a' and 'c'. This is a critical step in verifying that the chosen factors are correct for the factorization of the quadratic equation.
💡Solutions
Solutions to an equation are the values of the variable that satisfy the equation. In the context of the video, the solutions are the values of 'x' that make each binomial in the factorized form of the quadratic equation equal to zero. The script demonstrates how to find these solutions by setting each binomial equal to zero and solving for 'x'.
Highlights

Introduction to a simple method for factorizing quadratic equations.

Example provided: Factorizing the quadratic equation 2x^2 + 7x - 4 = 0.

Explanation of identifying coefficients a, b, and c in a quadratic equation.

Step-by-step guide on multiplying a and c to find a product.

Using the middle coefficient b to find two factors of the product that add up to b.

Illustration of changing the signs of the factors to simplify the equation.

Division of the factors by the leading coefficient a to find the values for x.

Second example with the equation 2x^2 - 2x - 12 = 0 and its factorization process.

Demonstration of finding factors that multiply to -24 and add up to -2.

Method of flipping the signs of the factors for the second equation.

Solving for x by dividing the factors by the coefficient a.

Introduction of a more complex example: -10x^2 + 13x + 3 = 0.

Caution about finding the correct factors for the complex example.

Explanation of the trick involved in factorizing the third example.

Final example with the equation 6x^2 - x - 2 = 0 and its solution.

Identification of factors that multiply to -12 and add up to -1.

Conclusion summarizing the simple method for factorizing quadratic equations.

Invitation for feedback and closing remarks of the video.

Transcripts

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