$p^2-10xp+16x^2-q^2+6xq
Answer (1)
Group the terms involving p and q separately.
Complete the square for the p terms and the q terms.
Recognize the expression as a difference of squares.
Apply the difference of squares factorization and simplify: ( p − q − 2 x ) ( p + q − 8 x ) .
Explanation
Problem Analysis We are given the expression p 2 − 10 x p + 16 x 2 − q 2 + 6 x q and our goal is to factor it. We will group the terms involving p and q separately and then complete the square for each group.
Grouping Terms First, let's group the terms: ( p 2 − 10 x p + 16 x 2 ) − ( q 2 − 6 x q ) .
Completing the Square for p Terms Now, complete the square for the p terms. We have p 2 − 10 x p . To complete the square, we need to add and subtract ( 10 x /2 ) 2 = ( 5 x ) 2 = 25 x 2 . So, we rewrite the p terms as: ( p 2 − 10 x p + 25 x 2 ) − 25 x 2 + 16 x 2 = ( p − 5 x ) 2 − 9 x 2 .
Completing the Square for q Terms Next, complete the square for the q terms. We have q 2 − 6 x q . To complete the square, we need to add and subtract ( 6 x /2 ) 2 = ( 3 x ) 2 = 9 x 2 . So, we rewrite the q terms as: q 2 − 6 x q + 9 x 2 − 9 x 2 = ( q − 3 x ) 2 − 9 x 2 . Therefore, − ( q 2 − 6 x q ) = − (( q − 3 x ) 2 − 9 x 2 ) = − ( q − 3 x ) 2 + 9 x 2 .
Substituting Back Now, substitute these back into the original expression: ( p − 5 x ) 2 − 9 x 2 − ( q − 3 x ) 2 + 9 x 2 .
Simplifying Simplify by cancelling the 9 x 2 terms: ( p − 5 x ) 2 − ( q − 3 x ) 2 .
Difference of Squares Recognize this as a difference of squares: A 2 − B 2 = ( A − B ) ( A + B ) , where A = ( p − 5 x ) and B = ( q − 3 x ) .
Applying Factorization Apply the difference of squares factorization: (( p − 5 x ) − ( q − 3 x )) (( p − 5 x ) + ( q − 3 x )) .
Simplifying Factors Simplify the factors: ( p − 5 x − q + 3 x ) ( p − 5 x + q − 3 x ) = ( p − q − 2 x ) ( p + q − 8 x ) .
Final Answer Therefore, the final factored form is ( p − q − 2 x ) ( p + q − 8 x ) .
Examples
Factoring expressions is a fundamental skill in algebra and is used extensively in various fields such as physics, engineering, and computer science. For example, in physics, you might encounter a situation where you need to find the equilibrium points of a system, which often involves solving equations that require factoring. Consider a simple model where the potential energy V of a particle depends on its position x as V ( x ) = x 4 − 5 x 2 + 4 . To find the equilibrium points, you need to find where the force (the derivative of the potential energy) is zero. This leads to the equation 4 x 3 − 10 x = 0 , which can be factored as 2 x ( 2 x 2 − 5 ) = 0 . From this factored form, you can easily find the equilibrium points x = 0 and x = ± 2 5 .
