Solving Quadratic Equations With Exponents

The Organic Chemistry Tutor
11 Jan 202007:12
EducationalLearning
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TLDRThe video script provides a detailed walkthrough of solving a complex algebraic equation: (2x^2 - 7x + 3)^4 = 2401. The presenter first simplifies the equation by taking the fourth root of both sides, leading to a quadratic equation. They then attempt to factorize the quadratic, identifying the correct pair of numbers that add up to -7 and multiply to -8. Through factoring by grouping, they derive two possible solutions: x = 4 and x = -1/2. The presenter verifies both solutions by substituting them back into the original equation, confirming their correctness. The script concludes with a clear explanation of the process and the final answers.

Takeaways
  • 🧐 The problem involves solving a complex equation where \((2x^2 - 7x + 3)^4 = 2401\).
  • 📚 The first step is to simplify the equation by taking the fourth root of both sides to eliminate the exponent.
  • 🔢 The fourth root of 2401 is found to be 7, as \(7^4 = 2401\).
  • ✂️ After simplification, the equation becomes \(2x^2 - 7x - 4 = 0\).
  • 🔍 The equation is now a quadratic one, which can be solved by factoring or using the quadratic formula.
  • 📐 The decision is made to attempt factoring the quadratic equation rather than using the quadratic formula.
  • 🔑 The process involves finding two numbers that multiply to \(2 imes -4 = -8\) and add up to \(-7\), which are \(-8\) and \(1\).
  • 📝 Factoring by grouping is used, which requires taking out the greatest common factor (GCF) from the terms.
  • 🌟 The equation factors into \((x - 4)(2x + 1) = 0\) after rearranging and factoring out the GCFs.
  • ✅ The zero product property is applied to find the solutions: \(x = 4\) and \(x = -\frac{1}{2}\).
  • 🔍 Both solutions are verified by substituting them back into the original equation to ensure they satisfy it.
Q & A

  • What is the original equation presented in the video?

    -The original equation is (2x^2 - 7x + 3)^4 = 2401.

  • What is the first step suggested to simplify the equation?

    -The first step is to get rid of the exponent by taking the fourth root of both sides of the equation.

  • What is the result of taking the fourth root of 2401?

    -The fourth root of 2401 is 7, as 7^4 equals 2401.

  • How is the equation transformed after taking the fourth root?

    -The equation becomes 2x^2 - 7x + 3 = 7 after taking the fourth root.

  • What operation is performed next to simplify the equation further?

    -The next operation is to subtract 7 from both sides, resulting in a quadratic equation 2x^2 - 7x - 4 = 0.

  • Why is factoring by grouping suggested to solve the quadratic equation?

    -Factoring by grouping is suggested because it can simplify the process of finding the roots of the quadratic equation.

  • What are the two numbers that multiply to -8 and add to -7?

    -The two numbers are -8 and +1, as (-8) * 1 = -8 and -8 + 1 = -7.

  • How is the quadratic equation factored using the numbers found?

    -The equation is factored as (x - 4)(2x + 1) = 0 by using the numbers -8 and +1.

  • What are the two possible solutions for x from the factored equation?

    -The two possible solutions for x are x = 4 and x = -1/2.

  • How is the correctness of the solutions verified?

    -The correctness of the solutions is verified by plugging the values of x back into the original equation and checking if both sides are equal.

  • What is the conclusion about the solutions after verification?

    -Both x = 4 and x = -1/2 are correct solutions to the original equation.

Outlines
00:00
🔍 Problem Introduction and Initial Steps

The video begins with a complex algebraic problem involving an expression raised to the fourth power equaling 2401. The presenter suggests the first step is to simplify the equation by taking the fourth root of both sides to eliminate the exponent, resulting in a quadratic equation. The right side simplifies to 7, as 7 to the fourth power equals 2401. The presenter then guides the viewer through the process of transforming the equation into a standard quadratic form by subtracting 7 from both sides.

05:02
🧩 Factoring the Quadratic Equation

After simplifying the equation to a quadratic form, the presenter explores the possibility of factoring the equation rather than using the quadratic formula. The equation is manipulated to have a leading coefficient of 2 and a constant term of -4. The presenter looks for two numbers that multiply to -8 and add up to -7, eventually choosing 1 and -8. The equation is then rewritten with these terms and factored by grouping, leading to the factored form (x - 4)(2x + 1) = 0.

🎯 Solving for x Using the Zero Product Property

The presenter applies the zero product property to solve for x, setting each factor equal to zero and solving the resulting simpler equations. This yields two potential solutions: x = 4 and x = -1/2. The presenter then validates these solutions by substituting them back into the original equation to ensure they satisfy the equation, confirming that both solutions are correct.

📝 Conclusion and Verification

The video concludes with the presenter verifying the correctness of both solutions, x = 4 and x = -1/2, by substituting these values back into the original equation. The calculations confirm that both values indeed satisfy the equation, thus completing the problem-solving process. The presenter wraps up the video by summarizing the solutions and the steps taken to arrive at them.

Mindmap
Keywords
💡Exponentiation
Exponentiation is a mathematical operation where a number is raised to the power of another number. In the video, the concept is central to the problem where an expression is raised to the fourth power, indicating that the expression is multiplied by itself four times. The script discusses simplifying the equation by taking the fourth root of both sides to eliminate the exponent.
💡Quadratic Equation
A quadratic equation is a polynomial equation of degree two, typically in the form of ax^2 + bx + c = 0. In the video, after simplifying the original problem, the script arrives at a quadratic equation, which is then factored to find the values of x. The quadratic equation is a key part of the problem-solving process in the video.
💡Root
In mathematics, a root of an equation is the value that, when substituted into the equation, makes the equation true. The script mentions finding the fourth root of 2401 to simplify the original equation. The concept of roots is crucial for transforming the original problem into a more manageable form.
💡Factoring
Factoring is the process of breaking down a polynomial into a product of its factors. The video script describes factoring by grouping as a method to solve the quadratic equation. It involves rearranging terms and factoring out common factors to simplify the equation into a product of binomials.
💡Greatest Common Factor (GCF)
The GCF of two or more numbers is the largest number that divides evenly into each of the numbers. In the script, the GCF is used to simplify terms during the factoring process, helping to break down the quadratic equation into simpler components that can be more easily solved.
💡Zero Product Property
The zero product property states that if the product of two numbers is zero, then at least one of the numbers must be zero. In the video, this property is used to set each factor of the factored quadratic equation equal to zero, allowing for the determination of the possible values of x.
💡Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and multiplication, and non-negative integer exponents. The original problem in the video involves a polynomial raised to a power, and the script discusses simplifying and solving a resulting polynomial equation.
💡Coefficient
In algebra, a coefficient is a numerical factor in a term of a polynomial or an algebraic expression. The script refers to the leading coefficient, which is the coefficient of the term with the highest power of x, and it plays a role in determining the process of solving the quadratic equation.
💡Scientific Calculator
A scientific calculator is an electronic device used to perform more complex mathematical operations, including exponentiation and roots. The script suggests using a scientific calculator to find the fourth root of 2401, demonstrating the practical application of technology in mathematical problem-solving.
💡Verification
Verification in mathematics involves checking the correctness of a solution by substituting it back into the original equation. The video script concludes with the verification of the solutions found for x, ensuring that they satisfy the original equation, which is an important step in problem-solving.
Highlights

The problem involves solving a complex equation where (2x^2 - 7x + 3)^4 = 2401.

The first step is to eliminate the fourth power by taking the fourth root of both sides.

After simplification, the equation becomes 2x^2 - 7x + 3 = 7.

2401 is a perfect fourth power, specifically 7^4.

Subtract 7 from both sides to form a quadratic equation 2x^2 - 7x - 4 = 0.

The equation is now a quadratic that can be solved using factoring or the quadratic formula.

The decision is made to attempt factoring the quadratic equation.

The product of the leading coefficient and the constant term is calculated as -8.

The search is for two numbers that multiply to -8 and add up to -7.

The numbers 1 and -8 are identified as the correct pair for factoring.

The equation is rewritten with -8x + x to facilitate factoring by grouping.

Factoring by grouping is applied to the equation.

The greatest common factor (GCF) is extracted from the terms, resulting in (x - 4)(2x + 1) = 0.

The zero product property is used to set each factor equal to zero.

Solving the equations x - 4 = 0 and 2x + 1 = 0 yields x = 4 and x = -1/2.

The solutions are checked by substituting them back into the original equation.

Both x = 4 and x = -1/2 are verified to be correct solutions.

Transcripts

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