https://r2.erweima.ai/mathbot/image/17257804576734y3gdnbr.jpg

The image contains three mathematical problems, each requiring the determination of the roots of different functions. Here's a transcription and explanation for each problem:

---

**1.** Determine the real roots of \( f(x) = -0.5x^2 + 2.5x + 4.5 \).  
Use the quadratic formula and then the bisection method to locate the roots in the interval \( x_1 < x_2 \), where \( -1 \leq x_1 \leq 1 \) and \( 6 \leq x_2 \leq 7 \).

---

**2.** Determine the real roots of \( f(x) = 5x^2 - 5x^3 + 0x - 2 \).  
Use the bisection method to locate the roots in the intervals \( x_1 = 0 \) and \( x_2 = 1 \).

---

**3.** Determine the real roots of \( f(x) = -25x^4 - 90x^3 - 14x^2 - 8x + 0.75 \).  
Use the bisection method to locate the roots with \( x_1 = 4 \) and \( x_2 = 5 \).

---

### Solution Strategy:
1. **Quadratic Formula**: 
   For \( ax^2 + bx + c = 0 \), the quadratic formula is:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Use this for the first problem to find the real roots.

2. **Bisection Method**:
   The bisection method is an iterative numerical technique to find roots of a function by narrowing the interval where the function changes sign. The general steps:
   - Choose two initial points \( a \) and \( b \) such that \( f(a) \cdot f(b) < 0 \).
   - Compute the midpoint \( c = \frac{a+b}{2} \).
   - Evaluate \( f(c) \), and decide if the root lies between \( a \) and \( c \) or \( c \) and \( b \).
   - Repeat the process until the interval is sufficiently small.

Would you like me to start solving any of these problems in detail? 

### Follow-up Questions:
1. Would you like a detailed explanation of the quadratic formula for the first problem?
2. Should I demonstrate the bisection method for any of the intervals provided?
3. Are you familiar with how to derive the initial guesses for the bisection method?
4. Would you like me to graph any of the functions to visualize the roots?
5. Do you need a step-by-step breakdown of how to apply the bisection method manually?

#### Tip: 
Always check that the function changes sign over the chosen interval for the bisection method to ensure there is a root within that interval.

1. Determine the real roots of f(x) = -0.5x^2 + 2.5x + 4.5. Use the quadratic formula and then the bisection method to locate the roots in the interval -1 ≤ x1 ≤ 1 and 6 ≤ x2 ≤ 7.
2. Determine the real roots of f(x) = 5x^2 - 5x^3 + 0x - 2. Use the bisection method to locate the roots with x1 = 0 and x2 = 1.
3. Determine the real roots of f(x) = -25x^4 - 90x^3 - 14x^2 - 8x + 0.75. Use the bisection method to locate the roots with x1 = 4 and x2 = 5.

Explore step-by-step solutions for finding the real roots of quadratic and polynomial functions using the quadratic formula and the bisection method. Learn key algebraic and numerical concepts ideal for high school and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation