Final answer:
Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula. Five quadratic equations are solved using these methods, leading to the identification of their roots.
Step-by-step explanation:
The student has asked for the roots of several quadratic equations. To find the roots, we can either factorize the equation, complete the square, or use the quadratic formula.
Here is the step-by-step process for each equation:
- a. f(x) = (x + 5)² + 6: This equation does not have real roots because the expression (x + 5)² is always non-negative, and when you add 6, it remains positive.
- b. f(x) = x² - 5x + 6: This can be factored into (x - 2)(x - 3), so the roots are x = 2 and x = 3.
- c. f(x) = x² - 16x + 64: This is a perfect square trinomial, factoring to (x - 8)², hence a single root at x = 8.
- d. f(x) = (x - 1)² - 25: Rewrite as (x - 1)² = 25 and take square roots to find that x = 1 ± 5, giving roots x = 6 and x = -4.
- e. f(x) = -(x - 2)² + 4: Set the equation equal to zero to get (x - 2)² = 4. Taking square roots gives x = 2 ± 2. The roots are x = 0 and x = 4.
In summary, quadratic equations may require different approaches depending on their form.
Recognizing perfect squares can also simplify the process before applying the quadratic formula or other methods.