Solving Quadratic Equations without CD: Better Direction?

In summary, the conversation discusses a problem that can be solved using Quadratics and grouping. The suggested approach is to set u = x + x^{-1} and use the equation u^3 + u^2 - 2u - 30 = (u- 3)(u^2 + 4u - 10) = 0 to find the roots. The other roots u = -2 \pm \sqrt{14} can also be used, but may lead to a more complicated equation.
  • #1
karush
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ok this was posted on LinkedIn and sure it has already be answered
but usually these types of problems are resolved by way too many steps
so just wanted to proceed with this without looking at previous attempts

my first reaction was to get a CD but would introduce a bigger problem
however with the exp i presume you could do this by Quadratics and grouping

or is there a better direction?
 
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  • #2
Set [itex]u = x + x^{-1}[/itex]. Then [itex]x^2 + x^{-2} = u^2 - 2[/itex] and [itex]x^3 + x^{-3} = u^3 - 3u[/itex] so that [tex]u^3 + u^2 - 2u - 30 = (u- 3)(u^2 + 4u - 10) = 0.[/tex] Then completing the square in [itex]x[/itex] gives [tex]
(2x - u)^2 = u^2 - 4[/tex] and the choice [itex]u = 3[/itex] leads directly to [tex]
(2x - 3)^2 = 5.[/tex] The other roots [itex]u = -2 \pm \sqrt{14}[/itex] lead to [tex]
x^2 + 2x \mp \sqrt{14}x + 1 = 0.[/tex]
 
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Related to Solving Quadratic Equations without CD: Better Direction?

1. How do I solve quadratic equations without using a CD?

There are several methods for solving quadratic equations without using a CD. One method is the quadratic formula, which involves plugging in the coefficients of the equation into the formula and solving for the roots. Another method is completing the square, where you manipulate the equation to create a perfect square trinomial and then solve for the roots. You can also use factoring to solve quadratic equations without a CD.

2. What is the best direction to take when solving quadratic equations without a CD?

The best direction to take when solving quadratic equations without a CD depends on the specific equation and your personal preference. Some people find the quadratic formula to be the most straightforward method, while others prefer completing the square or factoring. It is important to practice and become familiar with all three methods so you can choose the best direction for each individual equation.

3. Can I solve all quadratic equations without using a CD?

Yes, all quadratic equations can be solved without using a CD. The quadratic formula, completing the square, and factoring are all general methods that can be applied to any quadratic equation. However, some equations may have complex solutions that involve imaginary numbers, which may require additional knowledge and techniques to solve.

4. Are there any advantages to solving quadratic equations without a CD?

Yes, there are several advantages to solving quadratic equations without a CD. First, it allows you to solve equations without relying on a specific tool or technology. This can be helpful in situations where a CD or calculator is not available. Additionally, understanding the different methods for solving quadratic equations can improve your overall understanding of algebraic concepts.

5. How can I check my work when solving quadratic equations without a CD?

To check your work when solving quadratic equations without a CD, you can plug the solutions back into the original equation and see if they satisfy the equation. Another method is to graph the equation and see if the solutions correspond to the x-intercepts on the graph. You can also use the quadratic formula to double-check your solutions.

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