Solve x 2 10x 24 by completing the square – Solve x^2 + 10x + 24 by completing the square is a comprehensive guide that delves into the concept of completing the square to solve quadratic equations. This technique involves manipulating an equation into a perfect square trinomial, which can then be easily factored and solved.

This guide provides a step-by-step approach to completing the square, along with examples and applications in various fields.

Completing the square is a valuable tool for solving quadratic equations, and this guide offers a thorough exploration of its benefits, limitations, and variations. By understanding this technique, readers will gain a deeper understanding of quadratic equations and their applications in real-world scenarios.

Completing the Square: Definition and Overview

Completing the square is a technique used in algebra to solve quadratic equations. It involves manipulating the equation to transform it into a perfect square trinomial, which can then be easily factored and solved.

The general form of a quadratic equation is ax^2 + bx + c = 0. To complete the square, we need to add and subtract a term to the left-hand side of the equation that will make it a perfect square trinomial.

The term to be added is (b/2)^2.

Solving x^2 + 10x + 24 by Completing the Square

Let’s solve the equation x^2 + 10x + 24 = 0 using completing the square:

1. Add and subtract (b/2)^2 to the left-hand side: x^2 + 10x + (5)^2
   

   (5)^2 + 24 = 0

2. Simplify: (x + 5)^2
   

   1 = 0

3. Factor: (x + 5)^2 = 1
4. Take the square root of both sides: x + 5 = ±1
5. Solve for x: x =
   

   5 ± 1

6. Therefore, the solutions are x =
   • 4 or x =
   • 6.

Applications of Completing the Square

Completing the square has various applications in real-world problems:

• Projectile motion: Determining the trajectory of a projectile launched at a certain angle and velocity.
• Circuit analysis: Calculating the resonant frequency of an LC circuit.
• Optimization problems: Finding the maximum or minimum value of a quadratic function.

Variations of Completing the Square, Solve x 2 10x 24 by completing the square

There are other methods related to completing the square:

• Vertex form: Expressing a quadratic equation in the form y = a(x – h)^2 + k, where (h, k) is the vertex of the parabola.
• Discriminant: Using the discriminant (b^2 – 4ac) to determine the nature of the solutions to a quadratic equation.

Advanced Techniques

Advanced techniques for completing the square include:

• Using complex numbers: Solving quadratic equations with complex solutions.
• Solving higher-order equations: Applying completing the square to equations of degree greater than 2.

FAQs: Solve X 2 10x 24 By Completing The Square

What is completing the square?

Completing the square is a technique used to solve quadratic equations by converting them into a perfect square trinomial, which can then be easily factored and solved.

When should I use completing the square?

Completing the square is particularly useful for solving quadratic equations that cannot be easily factored using other methods.

What are the benefits of completing the square?

Completing the square provides a systematic and straightforward approach to solving quadratic equations, making it a valuable tool for students and professionals alike.