Prove by showing a valid mathematical argument:
For what values of p and q would (p+q) ² = p²+q³
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Prove by showing a valid mathematical argument:
For what values of p and q would (p+q) ² = p²+q³
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Answer:
The values of p and q that satisfy the equation (p + q)² = p² + q³ are:
1) For q = 0, p can be any real number.
2) For q ≠ 0, p must be 0.
Step-by-step explanation:
To find the values of p and q that satisfy the equation (p + q)² = p² + q³, we'll start by expanding both sides of the equation and simplifying:
Expanding the left side:
(p + q)² = (p + q) * (p + q) = p² + 2pq + q²
Expanding the right side:
p² + q³
Now we can equate the two sides of the equation and solve for p and q:
p² + 2pq + q² = p² + q³
Subtracting p² from both sides:
2pq + q² = q³
Subtracting q³ from both sides:
2pq + q² - q³ = 0
Now we have a polynomial equation in terms of p and q. To find the values of p and q, we need to solve this equation. However, it is not possible to solve this equation analytically because it is a higher-order polynomial equation.
To proceed, we can analyze the equation and make some observations:
1) If q = 0, then the equation becomes 2pq = 0, which means p can be any real number.
2) If q ≠ 0, we can divide both sides of the equation by q to simplify it:
2p + q - q²/q = 0
2p + q - q = 0
2p = 0
p = 0
So, if q ≠ 0, then p must be 0.
In conclusion, the values of p and q that satisfy the equation (p + q)² = p² + q³ are:
1) For q = 0, p can be any real number.
2) For q ≠ 0, p must be 0.