Which of the following equations are equivalent to ax2+bx+c=0?

x2+bx=−c


ax2+bx=−c
(x+b/2a)2=b2−4ac2a
x=−b±√b2−4ac2a
(x+b/2a)2=b2−4ac4a2
x2+b/ax+c/a=0
x=−b±√b2−4ac2a

Let's analyze which of the provided equations are equivalent to the standard quadratic equation \( ax^2 + bx + c = 0 \).

1. **\( x^2 + bx = -c \)**

   Rearranging this equation:
   \[
   x^2 + bx + c = 0
   \]
   This matches the given equation if \( a = 1 \), so it is **not equivalent** in the general case when \( a \neq 1 \).

2. **\( ax^2 + bx = -c \)**

   Rearranging this:
   \[
   ax^2 + bx + c = 0
   \]
   This is **equivalent** to the quadratic equation \( ax^2 + bx + c = 0 \).

3. **\( \left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{2a} \)**

   This expression is incorrect. The correct form of the "completing the square" formula should be:
   \[
   \left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2}
   \]
   Hence, this is **not equivalent**.

4. **\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)**

   This is the quadratic formula, which is the solution of \( ax^2 + bx + c = 0 \), so it is **equivalent**.

5. **\( \left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2} \)**

   This is the correct form of completing the square and is derived from the quadratic equation \( ax^2 + bx + c = 0 \). So, it is **equivalent**.

6. **\( x^2 + \frac{b}{a}x + \frac{c}{a} = 0 \)**

   Dividing the original equation \( ax^2 + bx + c = 0 \) by \( a \), we get:
   \[
   x^2 + \frac{b}{a}x + \frac{c}{a} = 0
   \]
   So, this is **equivalent**.

7. **\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)**

   This is a repeat of equation 4, which is the quadratic formula, and it is **equivalent**.

### Summary of equivalent equations:
- **\( ax^2 + bx = -c \)**
- **\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)** (both occurrences)
- **\( \left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2} \)**
- **\( x^2 + \frac{b}{a}x + \frac{c}{a} = 0 \)**

Would you like further details on any of these transformations or the process?

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Here are 5 related questions:
1. How do you derive the quadratic formula from \( ax^2 + bx + c = 0 \)?
2. What is the process of completing the square for a quadratic equation?
3. Can the quadratic formula be used for equations without real solutions?
4. What does the discriminant \( \Delta = b^2 - 4ac \) tell us about the solutions?
5. How would the solutions change if \( a \), \( b \), or \( c \) are complex numbers?

**Tip:** Completing the square is a useful method to derive the quadratic formula and to solve quadratic equations manually!

Which of the following equations are equivalent to ax^2+bx+c=0?

x^2+bx=−c

ax^2+bx=−c

(x+b/2a)^2=b^2−4ac2a

x=−b±√b^2−4ac2a

(x+b/2a)^2=b^2−4ac4a^2

x^2+b/ax+c/a=0

x=−b±√b^2−4ac2a

Explore which equations are equivalent to the quadratic equation ax^2 + bx + c = 0, including the quadratic formula, completing the square, and factored forms. This lesson covers key algebraic techniques, perfect for students in Grades 9-12.

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