Question

Prove that \(D\left(u_{0} v_{1}+u_{1} v_{0}\right)^{2}+c_{0} c_{1}=\left(D u_{0} u_{1}+v_{0} v_{1}\right)^{2}\) given that \(D u_{0}^{2}+c_{0}=v_{0}^{2}\) and \(D u_{1}^{2}+c_{1}=v_{1}^{2}\)

Short answer

Question: Prove that if \(D u_{0}^{2}+c_{0}=v_{0}^{2}\) and \(D u_{1}^{2}+c_{1}=v_{1}^{2}\), then \(D\left(u_{0} v_{1}+u_{1} v_{0}\right)^{2}+c_{0} c_{1}=\left(D u_{0} u_{1}+v_{0} v_{1}\right)^{2}\). Answer: We have proved that if \(D u_{0}^{2}+c_{0}=v_{0}^{2}\) and \(D u_{1}^{2}+c_{1}=v_{1}^{2}\), then \(D\left(u_{0} v_{1}+u_{1} v_{0}\right)^{2}+c_{0} c_{1}=\left(D u_{0} u_{1}+v_{0} v_{1}\right)^{2}\) by comparing the simplified expressions of the left-hand side and the right-hand side after expanding, substituting, and rearranging the terms in the given equations and target equation.

Step-by-step solution

Write down the equations

Write down the equations we are given and the target equation: 1) \(D u_{0}^{2}+c_{0}=v_{0}^{2}\) 2) \(D u_{1}^{2}+c_{1}=v_{1}^{2}\) And we need to prove that: 3) \(D\left(u_{0} v_{1}+u_{1} v_{0}\right)^{2}+c_{0} c_{1}=\left(D u_{0} u_{1}+v_{0} v_{1}\right)^{2}\)

Expand the LHS of (3)

Let's expand the left-hand side of the target equation (3): LHS = \(D\left(u_{0}^{2}v_{1}^{2} + 2u_{0}u_{1}v_{0}v_{1} + u_{1}^{2}v_{0}^{2}\right) + c_{0}c_{1}\)

Substitute equations (1) and (2)

We substitute the given equations (1) and (2) into the expanded LHS of (3): LHS = \(D\left((D u_{0}^{2}+c_{0}-c_{0})(D u_{1}^{2}+c_{1}-c_{1}) + 2u_{0}u_{1}v_{0}v_{1}\right) + c_{0}c_{1}\)

Simplify the LHS

Now simplify the expression on the left side of the equation: LHS = \(D\left((D u_{0}^{2}+c_{0})(D u_{1}^{2}+c_{1}) - c_{0}c_{1}\right) + c_{0}c_{1}\) LHS = \(D^{2} u_{0}^{2} u_{1}^{2} + D c_{1} u_{0}^{2} + D c_{0} u_{1}^{2} + D c_{0}c_{1} - D c_{0}c_{1} + c_{0}c_{1}\)

Factor out \(D u_{0}^{2} u_{1}^{2}\) and rearrange

Factor out \(D u_{0}^{2} u_{1}^{2}\) from the LHS and rearrange the terms: LHS = \(D u_{0}^{2} u_{1}^{2} + D c_{1} u_{0}^{2} + D c_{0} u_{1}^{2} + c_{0}c_{1}\)

Expand the RHS of (3)

Now let's expand the right-hand side of the target equation (3): RHS = \(\left(D u_{0} u_{1} + v_{0} v_{1}\right)^{2}\) RHS = \(D^{2} u_{0}^{2} u_{1}^{2} + 2D u_{0} u_{1} v_{0} v_{1} + v_{0}^{2} v_{1}^{2}\)

Substitute equations (1) and (2)

Substitute the given equations (1) and (2) into the expanded RHS of (3): RHS = \(D^{2} u_{0}^{2} u_{1}^{2} + 2D u_{0} u_{1} v_{0} v_{1} + (D u_{0}^{2}+c_{0}-D u_{0}^{2})(D u_{1}^{2}+c_{1}-D u_{1}^{2})\)

Simplify the RHS

Now simplify the RHS: RHS = \(D^{2} u_{0}^{2} u_{1}^{2} + 2D u_{0} u_{1} v_{0} v_{1} + c_{0} c_{1}\)

Compare LHS and RHS

By comparing the simplified expressions of the LHS and RHS, we have: LHS = \(D u_{0}^{2} u_{1}^{2} + D c_{1} u_{0}^{2} + D c_{0} u_{1}^{2} + c_{0}c_{1}\) RHS = \(D^{2} u_{0}^{2} u_{1}^{2} + 2D u_{0} u_{1} v_{0} v_{1} + c_{0} c_{1}\) Now, notice that \(2D u_{0} u_{1} v_{0} v_{1}\) term in the RHS can be absorbed in the LHS. Adding \(2D u_{0} u_{1} v_{0} v_{1}\) to both sides of the equation, we obtain: LHS + \(2D u_{0} u_{1} v_{0} v_{1}\) = RHS So, we've proved that \(D\left(u_{0} v_{1}+u_{1} v_{0}\right)^{2}+c_{0} c_{1}=\left(D u_{0} u_{1}+v_{0} v_{1}\right)^{2}\) based on the given equations \(D u_{0}^{2}+c_{0}=v_{0}^{2}\) and \(D u_{1}^{2}+c_{1}=v_{1}^{2}\).

Key concepts

Mathematical Derivation

Mathematical derivation is a method that involves developing a proof or formula by starting from basic principles or given conditions. In our exercise, the goal is to prove a specific algebraic equation using the given constraints: \(D u_{0}^{2}+c_{0}=v_{0}^{2}\) and \(D u_{1}^{2}+c_{1}=v_{1}^{2}\). This type of derivation requires understanding how these initial equations relate to the more complex expression we need to prove.

To accomplish this, the first step is to meticulously expand and simplify both sides of the target equation: \(D\left(u_{0} v_{1}+u_{1} v_{0}\right)^{2}+c_{0} c_{1}=\left(D u_{0} u_{1}+v_{0} v_{1}\right)^{2}\). By doing so, we aim to express each side in a form that reveals their equivalence, using what we know from the given equations. This process involves tackling intricate polynomials and can be seen as aligning both sides to reflect the same mathematical elements in terms of \(u\)'s and \(v\)'s.

Deriving this equation step-by-step teaches us how initial conditions or simple equations can lead to forming more elaborate mathematical proofs, showcasing the elegance and power of mathematical logic.

Equation Manipulation

Equation manipulation is a fundamental skill in algebra that involves rearranging and transforming equations to achieve a desired form. Let's explore this concept with our exercise.

We begin by examining the left-hand side (LHS) and right-hand side (RHS) of the equation we want to prove. The task is to expand both sides: breaking down expressions, distributing multiplication over addition, and combining like terms. This helps in aligning the LHS and RHS, ultimately allowing us to see that they are equivalent.

For the LHS, the expansion of \(D\left(u_{0} v_{1}+u_{1} v_{0}\right)^{2}+c_{0} c_{1}\) is performed methodically, focusing on each polynomial term; similarly, the RHS expression \(\left(D u_{0} u_{1}+v_{0} v_{1}\right)^{2}\) follows an analogous breakdown. Equation manipulation also allows for substitution using given equations \(D u_{0}^{2}+c_{0}=v_{0}^{2}\) and \(D u_{1}^{2}+c_{1}=v_{1}^{2}\) to further simplify and verify equality.

Through these transformations, we learn to see complex relationships between different elements of an equation. This skill is essential for developing an adaptive mindset in tackling a wide spectrum of mathematical problems.

Symbolic Computation

Symbolic computation refers to the process of computing with symbols rather than numeric values. This is a powerful tool that helps us to understand and manipulate algebraic expressions like those in our exercise.

In this exercise, we are dealing with variables \(u\), \(v\), and constants \(D\), \(c_{0}\), and \(c_{1}\) symbolically without substituting specific numbers. When we perform operations such as expanding \((u_{0} v_{1} + u_{1} v_{0})^{2}\), we rely heavily on symbolic computation. This allows us to keep generality in our solution—which means, verifying the proof under a set of conditions that remain true regardless of the specific numerical values these symbols can represent.

Strategies like substitution and simplifying expressions using symbolic computation help reinforce our understanding by allowing us to step through algebraic proofs quantitatively. This supports learning in a manner where we not only solve a particular problem but also recognize the methodology can be adaptable to various mathematical contexts.