Solving Quadratic Equations With Exponents
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
- π§ The problem involves solving a complex equation where \((2x^2 - 7x + 3)^4 = 2401\).
- π The first step is to simplify the equation by taking the fourth root of both sides to eliminate the exponent.
- π’ The fourth root of 2401 is found to be 7, as \(7^4 = 2401\).
- βοΈ After simplification, the equation becomes \(2x^2 - 7x - 4 = 0\).
- π The equation is now a quadratic one, which can be solved by factoring or using the quadratic formula.
- π The decision is made to attempt factoring the quadratic equation rather than using the quadratic formula.
- π The process involves finding two numbers that multiply to \(2 imes -4 = -8\) and add up to \(-7\), which are \(-8\) and \(1\).
- π Factoring by grouping is used, which requires taking out the greatest common factor (GCF) from the terms.
- π The equation factors into \((x - 4)(2x + 1) = 0\) after rearranging and factoring out the GCFs.
- β The zero product property is applied to find the solutions: \(x = 4\) and \(x = -\frac{1}{2}\).
- π 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
π 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.
𧩠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.
π― 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.
π 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
π‘Exponentiation
π‘Quadratic Equation
π‘Root
π‘Factoring
π‘Greatest Common Factor (GCF)
π‘Zero Product Property
π‘Polynomial
π‘Coefficient
π‘Scientific Calculator
π‘Verification
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
Browse More Related Video
Solving Exponential Equations Using Logarithms & The Quadratic Formula
Exponential Logarithmic Equations
Solving a quadratic equation by factoring | Algebra II | Khan Academy
Radical equation | Radical simplification | math tricks |Math Olympiad | unique math tricks
Exponential Equations - Algebra and Precalculus
Can you solve this? | Exponential Equation | Algebra Problem.
5.0 / 5 (0 votes)
Thanks for rating: