To prove that the expressions (x+7)(x−3) and (x−3)(x+7) are equivalent using coordinate geometry, you can follow these steps for each of the options:
a. ) Prove the slopes are the same:
- Consider the equation y = (x+7)(x−3).
- Rewrite the equation in slope-intercept form (y = mx + b).
- Compare the slope (m) of this equation with the slope of the equation obtained by expanding (x−3)(x+7).
- If the slopes are the same, you have proved that the expressions are equivalent.
b. ) Prove the slopes are opposite reciprocals:
- Consider the equation y = (x+7)(x−3).
- Rewrite the equation in slope-intercept form (y = mx + b).
- Compare the slope (m) of this equation with the slope of the equation obtained by expanding (x−3)(x+7).
- If the slopes are opposite reciprocals, you have proved that the expressions are equivalent.
c. ) Prove the lengths are the same:
- Consider the two expressions (x+7)(x−3) and (x−3)(x+7).
- Treat them as quadratic functions and find their roots (x-intercepts) by setting y = 0.
- Calculate the distance between the roots for both expressions.
- If the distances are the same, you have proved that the expressions are equivalent.
d. ) Prove the midpoints are the same:
- Consider the two expressions (x+7)(x−3) and (x−3)(x+7).
- Treat them as quadratic functions and find the x-coordinate of their vertex (midpoint of roots).
- Calculate the vertex for both expressions.
- If the x-coordinates of the vertices are the same, you have proved that the expressions are equivalent.
Please Note: In each case, you would use coordinate geometry principles to compare the properties of the two expressions and show that they are equivalent. The specific method you choose will depend on which option (a, b, c, or d) you are trying to prove.