$\frac{(a − b + c)^2 – (a – b – c)^2}{b-a}$ is equal to :

Q: $\frac{(a − b + c)^2 – (a – b – c)^2}{b-a}$ is equal to :

To solve the expression $\frac{(a - b + c)^2 - (a - b - c)^2}{b-a}$ and verify why Option 3, $-4c$, is the correct answer, let's go through the steps: 1. **Expand the Squares:** - $(a - b + c)^2 = (a - b)^2 + 2(a - b)c + c^2$ - $(a - b - c)^2 = (a - b)^2 - 2(a - b)c + c^2$ 2. **Subtract the Two Expressions:** \[ (a - b + c)^2 - (a - b - c)^2 = [(a - b)^2 + 2(a - b)c + c^2] - [(a - b)^2 - 2(a - b)c + c^2] \] Simplifying this, we get: \[ = 2(a - b)c + c^2 - (-2(a - b)c + c^2) \] \[ = 2(a - b)c + c^2 + 2(a - b)c - c^2 \] \[ = 4(a - b)c \] 3. **Divide by $b-a$:** \[ \frac{4(a - b)c}{b-a} = \frac{4c(a - b)}{b-a} \] Since $(a - b) = -(b - a)$, we have: \[ = \frac{4c(-1)(b-a)}{b-a} = -4c \] Thus, the expression simplifies to $-4c$, which confirms that Option 3 is indeed the correct answer. **Explanation for Incorrect Options:** 1. **Option 1: $b - c$** - The expression simplifies to $-4c$, not $b - c$. The presence of $b$ in the numerator is canceled out during the simplification process, leaving only terms involving $c$. 2. **Option 2: $c - a$** - The expression does not simplify to $c - a$. The algebraic manipulation shows that the terms involving $a$ and $b$ cancel out, leaving a multiple of $c$. 3. **Option 4: $2c$** - The simplification process results in $-4c$, not $2c$. The factor of 4 and the negative sign are crucial and arise from the algebraic expansion and simplification. Therefore, based on the detailed algebraic manipulation, Option 3, $-4c$, is the correct answer.

Question

Accepted Answer

To solve the expression $\frac{(a - b + c)^2 - (a - b - c)^2}{b-a}$ and verify why Option 3, $-4c$, is the correct answer, let's go through the steps:

1. **Expand the Squares:**

- $(a - b + c)^2 = (a - b)^2 + 2(a - b)c + c^2$
   - $(a - b - c)^2 = (a - b)^2 - 2(a - b)c + c^2$

2. **Subtract the Two Expressions:**

$$
   (a - b + c)^2 - (a - b - c)^2 = [(a - b)^2 + 2(a - b)c + c^2] - [(a - b)^2 - 2(a - b)c + c^2]
   $$

Simplifying this, we get:

$$
   = 2(a - b)c + c^2 - (-2(a - b)c + c^2)
   $$
   $$
   = 2(a - b)c + c^2 + 2(a - b)c - c^2
   $$
   $$
   = 4(a - b)c
   $$

3. **Divide by $b-a$:**

$$
   \frac{4(a - b)c}{b-a} = \frac{4c(a - b)}{b-a}
   $$

Since $(a - b) = -(b - a)$, we have:

$$
   = \frac{4c(-1)(b-a)}{b-a} = -4c
   $$

Thus, the expression simplifies to $-4c$, which confirms that Option 3 is indeed the correct answer.

**Explanation for Incorrect Options:**

1. **Option 1: $b - c$**

- The expression simplifies to $-4c$, not $b - c$. The presence of $b$ in the numerator is canceled out during the simplification process, leaving only terms involving $c$.

2. **Option 2: $c - a$**

- The expression does not simplify to $c - a$. The algebraic manipulation shows that the terms involving $a$ and $b$ cancel out, leaving a multiple of $c$.

3. **Option 4: $2c$**

- The simplification process results in $-4c$, not $2c$. The factor of 4 and the negative sign are crucial and arise from the algebraic expansion and simplification.

Therefore, based on the detailed algebraic manipulation, Option 3, $-4c$, is the correct answer.