Question
Use the Distributive Property to verify which expression is equivalent to
$(x-5)(x-3)$ . (1 point)
$3x^{2}-4x^{2}-3x-15$
$x^{2}-2x+15$
$x^{2}-8x+15$
$2x-2$
$(x-5)(x-3)$ . (1 point)
$3x^{2}-4x^{2}-3x-15$
$x^{2}-2x+15$
$x^{2}-8x+15$
$2x-2$
Answers
GPT-4o mini
To verify which expression is equivalent to \((x-5)(x-3)\), we will use the Distributive Property to expand the expression:
\[
(x-5)(x-3) = x(x-3) - 5(x-3)
\]
Expanding each term:
1. \(x(x - 3) = x^2 - 3x\)
2. \(-5(x - 3) = -5x + 15\)
Now, combine these results:
\[
x^2 - 3x - 5x + 15 = x^2 - 8x + 15
\]
Thus, the expansion gives us:
\[
(x-5)(x-3) = x^2 - 8x + 15
\]
Now, let’s compare \(x^2 - 8x + 15\) with the given options:
1. \(3x^{2}-4x^{2}-3x-15\) simplifies to \(-x^2 - 3x - 15\)
2. \(x^{2}-2x+15\) remains \(x^2 - 2x + 15\)
3. \(x^{2}-8x+15\) matches exactly \(x^2 - 8x + 15\)
4. \(2x-2\) remains \(2x - 2\)
The expression that is equivalent to \((x-5)(x-3)\) is:
\[
\boxed{x^{2}-8x+15}
\]
\[
(x-5)(x-3) = x(x-3) - 5(x-3)
\]
Expanding each term:
1. \(x(x - 3) = x^2 - 3x\)
2. \(-5(x - 3) = -5x + 15\)
Now, combine these results:
\[
x^2 - 3x - 5x + 15 = x^2 - 8x + 15
\]
Thus, the expansion gives us:
\[
(x-5)(x-3) = x^2 - 8x + 15
\]
Now, let’s compare \(x^2 - 8x + 15\) with the given options:
1. \(3x^{2}-4x^{2}-3x-15\) simplifies to \(-x^2 - 3x - 15\)
2. \(x^{2}-2x+15\) remains \(x^2 - 2x + 15\)
3. \(x^{2}-8x+15\) matches exactly \(x^2 - 8x + 15\)
4. \(2x-2\) remains \(2x - 2\)
The expression that is equivalent to \((x-5)(x-3)\) is:
\[
\boxed{x^{2}-8x+15}
\]
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