Solving a quadratic equation by factoring | Algebra II | Khan Academy

Khan Academy
23 Jun 201006:22
EducationalLearning
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TLDRThis educational video script guides viewers through solving a quadratic equation \( s^2 - 2s - 35 = 0 \) by factoring. It introduces the concept of factoring by grouping and demonstrates the process using the numbers 5 and -7, which sum to -2 and multiply to -35. The script then shows how to rearrange and factor the equation into \( (s + 5)(s - 7) = 0 \), leading to the solutions \( s = -5 \) or \( s = 7 \). Additionally, it highlights a shortcut for equations with a leading coefficient of 1, allowing for direct factoring into binomials.

Takeaways
  • 📚 The script introduces the process of solving a quadratic equation of the form s² - 2s - 35 = 0.
  • 🔍 It suggests factoring the left-hand side of the equation as the most efficient method for solving it, especially when the equation is set equal to zero.
  • 📈 The script explains the factoring by grouping method, which involves finding two numbers that sum to -2 and multiply to -35.
  • 🔑 It identifies the numbers 5 and -7 as the solution to this requirement, which are used to split the middle term of the quadratic equation.
  • 📝 The process involves rewriting the equation as s² + 5s - 7s - 35 = 0, then grouping terms to factor by common factors.
  • 🧩 After grouping, the script shows the factoring out of s and -7, resulting in s(s + 5) - 7(s + 5) = 0.
  • 🎯 The script emphasizes that the product of two terms being equal to zero implies that at least one of the terms must be zero.
  • 📉 By setting each factor equal to zero, the script derives two potential solutions for s: s + 5 = 0 and s - 7 = 0.
  • 📌 Solving these gives the solutions s = -5 and s = 7, which are verified to satisfy the original equation.
  • 🚀 The script also mentions a shortcut for equations with a leading coefficient of 1, allowing for direct factoring without the need for grouping.
  • 📝 Lastly, the script illustrates how to directly factor the equation into (s + 5)(s - 7) = 0, leading to the same solutions.
Q & A

  • What type of equation is being discussed in the script?

    -The script discusses a quadratic equation, which is a polynomial equation of degree 2.

  • Why is factoring the left-hand side recommended for solving the equation s^2 - 2s - 35 = 0?

    -Factoring the left-hand side is recommended because it simplifies the process of finding the values of 's' that satisfy the equation when it is explicitly set to zero.

  • What is the significance of the numbers 5 and -7 in the context of the script?

    -The numbers 5 and -7 are significant because they are the two numbers that add up to -2 (the coefficient of the linear term) and multiply to -35 (the constant term), which are needed for factoring the quadratic equation.

  • How does the script suggest splitting the middle term in the quadratic equation for factoring by grouping?

    -The script suggests splitting the middle term into +5s and -7s, which allows for factoring by grouping and identifying common factors.

  • What is the common factor extracted from the first two terms after splitting in the script?

    -The common factor extracted from the first two terms (s^2 + 5s) is 's', resulting in s(s + 5).

  • What is the common factor extracted from the second two terms after splitting in the script?

    -The common factor extracted from the second two terms (-7s - 35) is -7, resulting in -7(s + 5).

  • How does the script explain the concept of a product of two numbers being equal to zero?

    -The script explains that if the product of two numbers is equal to zero, then at least one of the numbers must be zero, which leads to the possible solutions for 's'.

  • What are the two possible solutions for 's' derived from the factored equation in the script?

    -The two possible solutions for 's' are s = -5 and s = 7, which are derived from setting each factor equal to zero.

  • How does the script verify the solutions for 's'?

    -The script verifies the solutions by substituting s = -5 and s = 7 back into the original equation and showing that both result in the equation equaling zero.

  • What is the shortcut mentioned in the script for factoring quadratic equations with a leading coefficient of 1?

    -The shortcut mentioned is to directly factor the quadratic equation into the product of two binomials, where the binomials' coefficients are the numbers that add up to the linear coefficient and multiply to the constant term.

Outlines
00:00
📚 Solving Quadratic Equations by Factoring

This paragraph introduces the process of solving a quadratic equation s^2 - 2s - 35 = 0 by factoring. It explains that the best approach is to factor the left-hand side when the equation is equal to zero. The method involves finding two numbers that sum to -2 and multiply to -35, which are identified as 5 and -7. The equation is then split and factored by grouping, leading to the factored form (s + 5)(s - 7) = 0. The solution to the equation is found by setting each factor equal to zero, resulting in two possible values for s: -5 and 7. The paragraph also verifies these solutions by substituting them back into the original equation.

05:03
🛠️ Factoring Quadratics with a Leading Coefficient of 1

The second paragraph discusses an alternative method for factoring quadratic equations when the leading coefficient is 1, which simplifies the process. It explains that instead of the two-step factoring by grouping, one can directly identify the two numbers that sum to the linear coefficient (-2) and multiply to the constant term (-35). Once these numbers are found (5 and -7), the quadratic can be factored directly into the product of two binomials: (s + 5)(s - 7). The paragraph emphasizes that this shortcut is a more efficient way to factor such equations and confirms that the result is the same as the one obtained through the grouping method.

Mindmap
Keywords
💡Quadratic Equation
A quadratic equation is a polynomial equation of degree 2, typically in the form of ax^2 + bx + c = 0, where a, b, and c are constants. In the video, the quadratic equation s^2 - 2s - 35 = 0 is the central problem that the instructor aims to solve, demonstrating the process of factoring to find the values of s.
💡Factoring
Factoring is the process of breaking down a complex expression into a product of simpler expressions. In the context of the video, the instructor uses factoring to solve the quadratic equation by finding two binomials that, when multiplied, give the original quadratic expression.
💡Grouping
Grouping is a technique used in factoring where terms are grouped together to facilitate the factoring process. In the script, the instructor groups terms s^2 + 5s and -7s - 35 to find common factors, which leads to the factored form of the quadratic equation.
💡Coefficient
A coefficient is a numerical factor in a term of an algebraic expression. In the video, the leading coefficient is 1, which simplifies the factoring process as it indicates that the quadratic can be directly factored into binomials without additional steps.
💡Product of Two Numbers
The product of two numbers refers to the result of multiplying them together. In the script, the instructor emphasizes that if the product of two numbers is zero, then at least one of the numbers must be zero, which is a key insight for solving the factored quadratic equation.
💡Zero Product Property
The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. The instructor uses this property to set each factor in the factored quadratic equation to zero, solving for the variable s.
💡Solving for a Variable
Solving for a variable involves finding the value(s) of the variable that make an equation true. In the video, the instructor solves for the variable s by setting each factor of the factored equation to zero and solving the resulting simpler equations.
💡Binomial
A binomial is an algebraic expression with two terms, typically in the form of ax + b. The script discusses factoring the quadratic equation into binomials, which are then set to zero to find the solutions for s.
💡Common Factor
A common factor is a factor that is present in each term of a group of terms. In the script, the instructor identifies s as a common factor in the grouped terms s^2 + 5s and uses it to factor out, simplifying the expression.
💡Shortcut
A shortcut is a quicker or more efficient method to achieve a result. The instructor mentions a shortcut for factoring when the leading coefficient is 1, allowing for direct factoring into binomials without the need for grouping.
💡Verification
Verification in mathematics involves checking the correctness of a solution by substituting the solution back into the original equation. The script includes verification of the solutions s = -5 and s = 7 by showing that they satisfy the original quadratic equation.
Highlights

Introduction to solving quadratic equations by factoring when the equation equals zero.

Explanation of the traditional algebraic method versus the factoring approach for quadratic equations.

Demonstration of factoring by grouping as a method to solve the given quadratic equation.

Identification of two numbers that sum to -2 and multiply to -35 for factoring.

The choice of 5 and -7 as factors that meet the criteria for the equation.

Splitting the middle term of the quadratic equation for factoring by grouping.

Factoring out the common terms of s and -7 from the grouped terms.

Revealing that both terms have a common factor of s+5, leading to further factoring.

Understanding the product of two numbers equating to zero implies at least one must be zero.

Solving for s by setting each factor equal to zero, resulting in s = -5 and s = 7.

Verification of the solutions by substituting back into the original equation.

Introduction of a shortcut for factoring when the leading coefficient is 1.

Explanation of the shortcut method using the pattern of x^2 + (a+b)x + ab.

Direct factoring into the product of binomials using the shortcut method.

Comparison between the factoring by grouping and the shortcut method for efficiency.

Emphasizing that both methods are valid for solving quadratic equations.

Transcripts

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