Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Solving Quadratic Equations: Guess and Check Method
**Equation:**
\[ x^2 - 3x - 18 \]
**Method: Guess and Check**
### Explanation:
To solve the quadratic equation using the guess and check method, you try to find two numbers that multiply to give -18 and add to give -3. This is useful for factoring quadratics that aren't easily simplified.
**Steps:**
1. **Identify coefficients:**
- The equation is in the form \( ax^2 + bx + c \).
- Here, \( a = 1 \), \( b = -3 \), \( c = -18 \).
2. **Find two numbers:**
- Look for two numbers that multiply to \( ac = 1 \times -18 = -18 \), and add up to \( b = -3 \).
3. **Test possible pairs:**
- Possible pairs for multiplication are:
- (-6, 3)
- (6, -3)
- (-9, 2)
- (9, -2)
- Among these, (-6, 3) satisfies both multiplying to -18 and adding to -3.
4. **Form factor pairs:**
- Using the pair (-6, 3), rewrite the middle term:
- \( x^2 - 6x + 3x - 18 \)
- Factor by grouping:
- \( x(x - 6) + 3(x - 6) \)
- Factor out the common term:
- \( (x + 3)(x - 6) \)
5. **Solve for \( x \):**
- Set each factor to zero:
- \( x + 3 = 0 \) or \( x - 6 = 0 \)
- Solutions are:
- \( x = -3 \) or \( x = 6 \)
In this method, guessing and checking different pairs helps find the solutions to the quadratic equation by transforming it into simpler linear equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc37dbff0-e053-476c-a940-a199db5d9106%2F89a6e738-0613-43bc-a23e-05a958874bba%2Fpljge9h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Quadratic Equations: Guess and Check Method
**Equation:**
\[ x^2 - 3x - 18 \]
**Method: Guess and Check**
### Explanation:
To solve the quadratic equation using the guess and check method, you try to find two numbers that multiply to give -18 and add to give -3. This is useful for factoring quadratics that aren't easily simplified.
**Steps:**
1. **Identify coefficients:**
- The equation is in the form \( ax^2 + bx + c \).
- Here, \( a = 1 \), \( b = -3 \), \( c = -18 \).
2. **Find two numbers:**
- Look for two numbers that multiply to \( ac = 1 \times -18 = -18 \), and add up to \( b = -3 \).
3. **Test possible pairs:**
- Possible pairs for multiplication are:
- (-6, 3)
- (6, -3)
- (-9, 2)
- (9, -2)
- Among these, (-6, 3) satisfies both multiplying to -18 and adding to -3.
4. **Form factor pairs:**
- Using the pair (-6, 3), rewrite the middle term:
- \( x^2 - 6x + 3x - 18 \)
- Factor by grouping:
- \( x(x - 6) + 3(x - 6) \)
- Factor out the common term:
- \( (x + 3)(x - 6) \)
5. **Solve for \( x \):**
- Set each factor to zero:
- \( x + 3 = 0 \) or \( x - 6 = 0 \)
- Solutions are:
- \( x = -3 \) or \( x = 6 \)
In this method, guessing and checking different pairs helps find the solutions to the quadratic equation by transforming it into simpler linear equations.
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