Question

1. solve: \\((x + 5)(x - 7) = 0\\)

\\(x = 35\\)

\\(x = -7\\) or \\(5\\)

\\(x = \sqrt{35}\\)

\\(x = -5\\) or \\(7\\)

Explanation:

Identify the given equation

We are given a quadratic equation that is already factored and set equal to zero:

$$(x + 5)(x - 7) = 0$$

Apply the Zero Product Property

The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be equal to zero.
For our equation, this means:

$$x + 5 = 0 \quad \text{or} \quad x - 7 = 0$$

Solve each linear equation

Now, we solve each simple linear equation for \(x\):

1. For the first factor:

$$x + 5 = 0 \implies x = -5$$

1. For the second factor:

$$x - 7 = 0 \implies x = 7$$

State the final solutions

Combining our results, the solutions to the equation are:

$$x = -5 \quad \text{or} \quad x = 7$$

Answer:

• (A) \(x = 35\)
• (B) \(x = -7 \text{ or } 5\)
• (C) \(x = \sqrt{35}\)
• (D) \(x = -5 \text{ or } 7\) (Correct answer)