How to Prove ( c - d^2 ge 0 ) and Its Applications in Algebraic Expressions

Mathematics is a field that relies heavily on logical reasoning and proof. One common type of proof involves inequalities and algebraic manipulations. In this article, we will explore how to prove the inequality ( c - d^2 ge 0 ) and its applications in more complex algebraic expressions. We will also provide a detailed example to demonstrate the use of this inequality in solving a specific problem.

Proving ( c - d^2 ge 0 )

The inequality ( c - d^2 ge 0 ) can be proven by rewriting the expression in a form that reveals its non-negativity. Here's the detailed proof:

Step 1: Start with the expression: ( c - d^2 ).

Step 2: Write it as a difference of squares: ( (c - d^2) c^2 - 2cd d^2 - c^2 d^2 ).

Step 3: Regroup the terms: ( c^2 - 2cd d^2 - (c^2 - d^2) ).

Step 4: Notice that ( c^2 - 2cd d^2 ) is a perfect square: ( (c - d)^2 ).

Step 5: Therefore, the inequality can be rewritten as: ( (c - d)^2 - (c^2 - d^2) ge 0 ).

Step 6: Simplify the expression: ( (c - d)^2 - (c - d)(c d) ge 0 ).

Step 7: Factor the expression: ( (c - d)^2 - (c - d)(c d) (c - d)[(c - d) - (c d)] ge 0 ).

Step 8: Simplify the brackets: ( (c - d)(-2d) ge 0 ).

Step 9: Since ( (c - d)(-2d) ge 0 ), it implies that ( c - d ge 0 ) or ( d ge 0 ), which confirms ( c - d^2 ge 0 ).

Applications in Algebraic Expressions

Now, let's apply this inequality to solve a more complex algebraic problem. Specifically, we will prove the inequality ( abcd le cdab ) if ( ab cd0 ), assuming ( ab e 0 ).

Step 1: Given ( ab cd0 ), we need to prove ( abcd le cdab ).

Step 2: Start with the expression: ( abcd - cdab ).

Step 3: Rearrange the terms: ( abcd - cdab abcd - cdab ).

Step 4: Write the expression in a different form: ( abcd - cdab abc^2d^2 - cdab^2 ).

Step 5: Factor out the common terms: ( abc^2d^2 - cdab^2 (abc^2d^2 - 2abcdb^2 (abcd - abcd) ).

Step 6: Use the inequality ( c - d^2 ge 0 ) to rewrite the expression: ( (abc^2d^2 - 2abcd) - (2abcd - abcd) ge 0 ).

Step 7: Simplify the expression: ( (abc^2d^2 - 2abcd) - (abcd - abcd) ge 0 ).

Step 8: This simplifies to: ( abc^2d^2 - 2abcd ge 0 ).

Step 9: Since ( abc^2d^2 ge 2abcd ), it follows that ( abcd le cdab ).

Conclusion

In conclusion, the inequality ( c - d^2 ge 0 ) can be proven using basic algebraic manipulations. This inequality has significant applications in solving more complex algebraic expressions, as demonstrated in the problem of proving ( abcd le cdab ) given ( ab cd0 ). Understanding and applying such inequalities can greatly enhance problem-solving skills in mathematics.