Solving the Equation: What is the Square of a Number Equal to 16 Times That Number?

Mathematics is a fascinating domain, particularly when it comes to solving equations. One such problem that often captures the attention of both students and math enthusiasts is the question: What is the square of a number equal to 16 times that number?

The equation in question is x^2 16x. This is a quadratic equation, a fundamental concept in algebra, and it offers valuable insights into how to solve for variables in equations involving squares and multiples of a variable. Let's explore the process in detail and provide the solutions step by step.

Step-by-Step Solution

Given the equation x^2 16x, the first step is to rearrange the equation so that all terms are on one side, resulting in:

Next, we factor out the common term, which in this case is x, to simplify the equation further:

Identifying the Solutions

The equation is now in a form that can be easily solved by setting each factor to zero:

Therefore, the equation x^2 16x has two solutions: x 0 and x 16.

Verification of Solutions

To ensure the accuracy of these solutions, we can verify them by substituting them back into the original equation.

Substituting x 0 into the original equation x^2 16x gives:

This is true, confirming that x 0 is a valid solution.

Substituting x 16 into the original equation x^2 16x gives:

This is also true, confirming that x 16 is another valid solution.

Conclusion

In conclusion, the equation x^2 16x has two solutions: x 0 and x 16. This problem demonstrates the importance of algebraic manipulation and factoring in solving quadratic equations, reinforcing the foundational skills in mathematics.

Understanding and solving such equations not only enhances one's mathematical prowess but also aids in tackling more complex problems in various fields that rely on algebra and equations.

Keywords: quadratic equation, solve for x, algebraic solutions