Solving Simultaneous Equations Through Completing the Square: A Detailed Guide

When dealing with quadratic equations in mathematics, there are several methods to find the roots or solutions of the equation. One such method is completing the square, which is particularly useful in solving quadratic equations that may not factor easily. In this guide, we will walk you through the steps of completing the square, using a specific example, and explain how it can be applied to solving simultaneous equations.

Understanding Simultaneous Equations

Simultaneous equations are a set of equations that share the same variables. To solve them, we need to find the values of the variables that satisfy all the equations in the set. One common method for solving these equations is through methods such as substitution, elimination, or, in the case of quadratic equations, completing the square.

The Completing the Square Method: A Step-by-Step Guide

The completing the square method is a powerful technique for solving quadratic equations. The goal is to convert a quadratic equation into a perfect square trinomial, which can then be easily solved. Let's walk through the steps using the example equation x2 9x 0.

Example: Solving x2 9x 0 Using Completing the Square

Step 1: Move the constant term to the right side of the equation.

x2 9x 0 becomes x2 9x 0 ?0.

Step 2: Take half of the coefficient of the linear term (9) and square it.

Half of 9 is 4.5, and (4.5)2 20.25.

Step 3: Add and subtract the square of half the coefficient of the linear term to the left side of the equation.

x2 9x 20.25 20.25, which is rewritten as (x 4.5)2 20.25.

Step 4: Take the square root of both sides of the equation, remembering to consider both the positive and negative roots.

x 4.5 ±4.5, which gives us two solutions: x 0 and x -9.

Step 5: Simplify the solutions to get the final values of x.

x -9 or x 0.

This method can be particularly helpful when dealing with simultaneous equations, where the same variable appears in more than one equation. By solving each equation individually using completing the square, we can then find the common solutions that satisfy all equations in the set.

Applying Completing the Square to Simultaneous Equations

When dealing with simultaneous equations, especially quadratic ones, the completing the square method can be a powerful tool. Here’s a scenario where we can apply this technique:

Equation 1: x2 9x 0 Equation 2: y2 - 5y 0

To solve these simultaneously, we can first solve each equation individually using the completing the square method.

Example of Simultaneous Equations

Equation 1: x2 9x 0

x2 9x 0 becomes x2 9x 0. The coefficient of x is 9, half of which is 4.5, and (4.5)2 20.25. Add and subtract 20.25 to get: x2 9x 20.25 20.25 or (x 4.5)2 20.25. Square root both sides: x 4.5 ±4.5, giving x 0 and x -9.

Equation 2: y2 - 5y 0

y2 - 5y 0 becomes y2 - 5y 0. The coefficient of y is -5, half of which is -2.5, and (-2.5)2 6.25. Add and subtract 6.25 to get: y2 - 5y 6.25 6.25 or (y - 2.5)2 6.25. Square root both sides: y - 2.5 ±2.5, giving y 0 and y 5.

Now, by comparing the solutions from both equations, we can find the common solutions that satisfy both. In this case, the solutions are x 0 and y 0.

Conclusion

Completing the square is a versatile and efficient method for solving quadratic equations, which can be particularly useful in solving simultaneous equations. By understanding this technique thoroughly, you can tackle complex equations with ease and accuracy. Whether you’re a student, a teacher, or a professional mathematician, mastering the completing the square method will significantly enhance your problem-solving capabilities.