Question

Use the distributive property to verify which expression is equivalent to (x-5)(x-3)

A. 3x^2-4x^2-3x-15
B. r^2-2x+15
C. 2x-2
D. r^2-8x+15

Answer 1

To verify which expression is equivalent to \((x-5)(x-3)\), we will first expand it using the distributive property (also known as the FOIL method for binomials).

\[
(x-5)(x-3) = x \cdot x + x \cdot (-3) - 5 \cdot x - 5 \cdot (-3)
\]

Now let's calculate each part:

1. \(x \cdot x = x^2\)
2. \(x \cdot (-3) = -3x\)
3. \(-5 \cdot x = -5x\)
4. \(-5 \cdot (-3) = +15\)

Now we can combine these results:

\[
x^2 - 3x - 5x + 15 = x^2 - 8x + 15
\]

So, the expression we derived from \((x - 5)(x - 3)\) is:

\[
x^2 - 8x + 15
\]

Now let's compare this with the options provided:

A. \(3x^2 - 4x^2 - 3x - 15\)

This can simplify to:

\[
(3x^2 - 4x^2) - 3x - 15 = -x^2 - 3x - 15 \quad \text{(not equivalent)}
\]

B. \(r^2 - 2x + 15\)

This includes \(r\) instead of \(x\) and does not match \(x^2 - 8x + 15\) \quad \text{(not equivalent)}.

C. \(2x - 2\)

This is a linear expression and not quadratic \quad \text{(not equivalent)}.

D. \(r^2 - 8x + 15\)

Similarly, this includes \(r\) instead of \(x\) and also does not match \quad \text{(not equivalent)}.

None of the options A, B, C, or D are equivalent to \(x^2 - 8x + 15\).

The correct equivalent expression is **not provided** among the choices. The correct equivalent expression obtained from the multiplication is \(x^2 - 8x + 15\).