Algebra II Week 4 Discussion

Algebra Ii Week 4 Discussionin This Discussion You Will Solve Quadr

Algebra II – Week 4 Discussion In this discussion, you will solve quadratic equations by two main methods: factoring and using the quadratic formula. Read the following instructions in order and view the example to complete this discussion. Please find the problems assigned to you in the table below: For the factoring problem, be sure you show all steps to the factoring and solving. Show a check of your solutions back into the original equation. For the Quadratic Formula problem, be sure that you use readable notation while you are working the computational steps.

Present your final solutions as decimal approximations carried out to the third decimal place. Due to the nature of these solutions, no check is required. Incorporate the following four math vocabulary words into your discussion. Use bold font to emphasize the words in your writing (Do not write definitions for the words; use them appropriately in sentences describing your math work.):

  • Quadratic formula
  • Factoring
  • Completing the square
  • Discriminant

Your initial post should be at least 250 words in length.

Paper For Above instruction

Quadratic equations are fundamental in algebra, representing parabolic relationships in various scientific and real-world contexts. In this discussion, I will demonstrate solving quadratic equations using two primary methods: factoring and the quadratic formula. These approaches are essential tools for algebraic problem-solving, each suitable for different types of quadratic equations based on their structure.

Let us first consider the factoring method. Consider the quadratic equation x² + 5x - 24 = 0. To solve by factoring, I look for two binomials whose product yields the original quadratic. The two numbers that multiply to -24 and add to 5 are 8 and -3. Therefore, I factor the quadratic as (x + 8)(x - 3) = 0. Setting each factor to zero gives solutions x = -8 and x = 3. To verify these solutions, I substitute back into the original equation: for x = -8, (-8)² + 5(-8) - 24 = 64 - 40 - 24 = 0; for x = 3, 9 + 15 - 24 = 0. Both solutions satisfy the original equation, confirming their correctness.

>Next, I apply the quadratic formula to solve the quadratic equation 2x² - 4x - 6 = 0. The quadratic formula states that x = [-b ± √(b² - 4ac)] / 2a, where a = 2, b = -4, and c = -6. Calculating the discriminant, Δ = b² - 4ac = 16 - 4(2)(-6) = 16 + 48 = 64. Since the discriminant is positive, there are two real solutions. Computing the solutions, x = [4 ± √64] / 4, which simplifies to x = [4 ± 8] / 4. This yields x = (4 + 8)/4 = 3 and x = (4 - 8)/4 = -1. These solutions are approximate to three decimal places, but since they are exact in this case, no further rounding is necessary.

Throughout these calculations, the importance of the discriminant becomes evident, as it informs the nature and number of solutions without performing all algebraic steps explicitly. The method of completing the square can also be employed for certain quadratics but is not necessary in this context. The quadratic formula provides a reliable means to find solutions when factoring is complex or not feasible, especially for quadratics with non-factorable roots.

In conclusion, mastering the methods of factoring and the quadratic formula enhances problem-solving skills in algebra. Understanding the discriminant aids in determining whether solutions will be real or complex, guiding the choice of method. These tools are fundamental in various applications, from physics to finance, where quadratic relationships are prevalent.

References

  • Blitzer, R. (2019). Algebra and Trigonometry (6th ed.). Pearson.
  • Anton, H., Bivens, I., & Davis, S. (2016). Algebra: A Combined Approach. Wiley.
  • Larson, R., Hostetler, R., & Edwards, B. (2019). Elementary Linear Algebra (8th ed.). Cengage Learning.
  • Aufmann, R. N., & Barker, V. C. (2018). College Algebra (8th ed.). Cengage Learning.
  • Swokowski, E. W., & Cole, J. A. (2018). Algebra and Trigonometry with Analytic Geometry. Cengage Learning.

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