rewrite each equation to standard form ax² + bx + c = 0, then identify the a b and c

1. 18x² + 9x = 7

2. y² + 15y = 8y

3. 20 - x² = x

4. 0 = p² - 7p

5. m² = 16

6. -25t² + 4 = 0

7. (x+3)² = 1

8. -3a = a² + 9/4

9. n² - 4 = 5 (n-3)

10. 5j (j - 3) - 4 = j - 6

Let's rewrite each equation to the standard form \( ax^2 + bx + c = 0 \), and then identify the coefficients \( a \), \( b \), and \( c \).

1. **Original equation:** \( 18x^2 + 9x = 7 \)

   **Standard form:** 
   \[
   18x^2 + 9x - 7 = 0
   \]
   **Coefficients:** \( a = 18 \), \( b = 9 \), \( c = -7 \)

2. **Original equation:** \( y^2 + 15y = 8y \)

   **Standard form:** 
   \[
   y^2 + 7y = 0
   \]
   **Coefficients:** \( a = 1 \), \( b = 7 \), \( c = 0 \)

3. **Original equation:** \( 20 - x^2 = x \)

   **Standard form:** 
   \[
   -x^2 - x + 20 = 0
   \]
   **Coefficients:** \( a = -1 \), \( b = -1 \), \( c = 20 \)

4. **Original equation:** \( 0 = p^2 - 7p \)

   **Standard form:** 
   \[
   p^2 - 7p = 0
   \]
   **Coefficients:** \( a = 1 \), \( b = -7 \), \( c = 0 \)

5. **Original equation:** \( m^2 = 16 \)

   **Standard form:** 
   \[
   m^2 - 16 = 0
   \]
   **Coefficients:** \( a = 1 \), \( b = 0 \), \( c = -16 \)

6. **Original equation:** \( -25t^2 + 4 = 0 \)

   **Standard form:** 
   \[
   -25t^2 + 4 = 0
   \]
   **Coefficients:** \( a = -25 \), \( b = 0 \), \( c = 4 \)

7. **Original equation:** \( (x+3)^2 = 1 \)

   **Standard form:**
   \[
   x^2 + 6x + 8 = 0
   \]
   **Coefficients:** \( a = 1 \), \( b = 6 \), \( c = 8 \)

8. **Original equation:** \( -3a = a^2 + \frac{9}{4} \)

   **Standard form:**
   \[
   a^2 + 3a + \frac{9}{4} = 0
   \]
   **Coefficients:** \( a = 1 \), \( b = 3 \), \( c = \frac{9}{4} \)

9. **Original equation:** \( n^2 - 4 = 5(n - 3) \)

   **Standard form:**
   \[
   n^2 - 9n + 11 = 0
   \]
   **Coefficients:** \( a = 1 \), \( b = -9 \), \( c = 11 \)

10. **Original equation:** \( 5j(j - 3) - 4 = j - 6 \)

    **Standard form:**
    \[
    5j^2 - 16j - 2 = 0
    \]
    **Coefficients:** \( a = 5 \), \( b = -16 \), \( c = -2 \)

Would you like more details on any of these transformations or have any questions?

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Here are five related questions:

1. How do you solve quadratic equations after converting them to standard form?
2. Can all quadratic equations be factored easily?
3. What are the methods to find the roots of a quadratic equation?
4. How does the discriminant help in understanding the nature of roots?
5. What are the applications of quadratic equations in real life?

**Tip:** When converting to standard form, always ensure all terms are on one side of the equation and combined properly before identifying \( a \), \( b \), and \( c \).

Learn how to rewrite various equations to the standard form ax^2 + bx + c = 0 and identify coefficients a, b, and c. Examples include solving equations involving quadratic expressions and transformations. Suitable for high school students.

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