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 p = 0 and q = 0 or q = 1.
Step-by-step explanation:
To find the values of p and q for which (p+q)² = p²+q³ holds true, we will set up the equation and solve for p and q.
Given: (p+q)² = p²+q³
Expanding the left side of the equation, we have:
p² + 2pq + q² = p² + q³
Subtracting p² from both sides, we get:
2pq + q² = q³
Next, let's focus on the right side of the equation:
2pq + q² = q³
Subtracting 2pq from both sides, we obtain:
q² - 2pq = q³ - 2pq
Factorizing q out from both terms on the left side, we have:
q(q - 2p) = q(q² - 2p)
Dividing both sides by q (while assuming q is not zero), we obtain:
q - 2p = q² - 2p
Rearranging terms, we get:
q² - q = 2p - 2p
Simplifying the right side, we have:
q² - q = 0
Now, factoring out q from both terms on the left side, we obtain:
q(q - 1) = 0
This equation holds true when either q = 0 or (q - 1) = 0.
1. If q = 0, substituting this value into the original equation (p+q)² = p²+q³, we have:
(p + 0)² = p² + 0³
p² = p²
This equation holds true for all values of p.
2. If q - 1 = 0, then q = 1. Substituting q = 1 into the original equation (p+q)² = p²+q³, we have:
(p + 1)² = p² + 1³
(p + 1)² = p² + 1
p² + 2p + 1 = p² + 1
Subtracting p² and 1 from both sides, we obtain:
2p = 0
Dividing both sides by 2 (assuming 2 is not zero), we have:
p = 0
Verified answer
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, dapat na 0 ang p.
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.