2x² + 9x +4=0 quadratic equations using factoring method or extracting square roots

To solve the quadratic equation \( 2x^2 + 9x + 4 = 0 \) using the factoring method, follow these steps:

### **Using the Factoring Method**

### Step 1: Multiply the coefficient of \( x^2 \) by the constant term.
- Multiply the coefficient of \( x^2 \) (which is 2) by the constant term (which is 4):

  \[
  2 \times 4 = 8
  \]

### Step 2: Find two numbers that multiply to 8 and add up to the coefficient of \( x \) (which is 9).
- We need to find two numbers that multiply to 8 and add to 9. These numbers are 8 and 1 because:

  \[
  8 \times 1 = 8
  \]

  \[
  8 + 1 = 9
  \]

### Step 3: Rewrite the middle term using the numbers found.
- Rewrite the equation by splitting the middle term (9x) into two terms using 8 and 1:

  \[
  2x^2 + 8x + x + 4 = 0
  \]

### Step 4: Factor by grouping.
- Group the terms into two pairs:

  \[
  (2x^2 + 8x) + (x + 4) = 0
  \]

- Factor out the common factors from each group:

  \[
  2x(x + 4) + 1(x + 4) = 0
  \]

### Step 5: Factor out the common binomial.
- Notice that \( (x + 4) \) is common in both terms:

  \[
  (2x + 1)(x + 4) = 0
  \]

### Step 6: Solve for \( x \).
- Set each factor equal to zero and solve for \( x \):

  1. \( 2x + 1 = 0 \)
     \[
     2x = -1 \implies x = \frac{-1}{2}
     \]

  2. \( x + 4 = 0 \)
     \[
     x = -4
     \]

### Final Answer:
The solutions to the equation \( 2x^2 + 9x + 4 = 0 \) are:

\[
x = \frac{-1}{2} \quad \text{or} \quad x = -4
\]

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### **Using the Square Roots Method**

The square roots method generally applies to equations that can be rewritten in the form \( ax^2 = c \). For this specific quadratic equation \( 2x^2 + 9x + 4 = 0 \), the square roots method is not straightforward, as the equation doesn't easily lend itself to being rewritten in that form. Therefore, the factoring method is the more appropriate technique in this case.

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### Follow-Up Questions:
1. How do you apply the quadratic formula to solve \( 2x^2 + 9x + 4 = 0 \)?
2. What is the significance of factoring in solving quadratic equations?
3. Can you provide another example of solving quadratic equations by factoring?
4. What happens if the quadratic equation cannot be factored easily?
5. How does completing the square differ from factoring?
6. When is it preferable to use the square roots method?
7. What role does the discriminant play in factoring quadratic equations?
8. How can you verify the solutions found by factoring?

### Tip:
If the factoring method doesn’t work, consider using the quadratic formula or completing the square as alternative methods to solve the quadratic equation.

Learn how to solve the quadratic equation 2x^2 + 9x + 4 using the factoring method with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring quadratic equations and applying the quadratic formula theorem. Suitable for students in Grades 9-12.

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