Math Problem Statement
Solution
Let's break down the two parts of this question.
Part (c): Rewrite in the form
This is asking us to complete the square.
- Start with:
- Factor out 2 from the first two terms:
- Complete the square inside the parentheses. Take half of 6, square it, and add/subtract it: So:
- Expand and simplify: Thus, the expression in completed square form is:
The parameters are:
- , , and .
Turning point:
Since the function is written as , the turning point is given by:
Part (d): Solve
This equation involves an absolute value. To solve it, we break it into two cases.
Case 1:
Case 2:
Thus, the solutions to the equation are:
Do you have any questions or would you like further clarification on these steps?
Here are 5 related questions to explore:
- How do you find the vertex of a quadratic function without completing the square?
- What does the value of indicate about the shape of a parabola?
- How would the solutions change if the absolute value equation in part (d) had a different constant on the right-hand side?
- Can a quadratic function have more than one turning point? Why or why not?
- How do you solve inequalities involving absolute values?
Tip: Always double-check your solutions when dealing with absolute value equations, as extraneous solutions can arise.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Completing the Square
Absolute Value Equations
Formulas
Standard form of a quadratic function f(x) = a(x + h)^2 + k
Absolute value equation |(x - 2) / (x + 3)| = 4
Theorems
Properties of quadratic functions
Absolute value properties
Suitable Grade Level
Grades 10-12