Solving Quadratic Equations A Comprehensive Guide

Jul 14, 2025 by Jeany 50 views

This article delves into the realm of quadratic equations, offering a comprehensive guide to solving equations of the form t^2 - 100 = 0, x^2 - 144 = 0, t^2 = 81, w^2 = 0, and 5p^2 + 64 = 0. Understanding quadratic equations is crucial in various fields, including mathematics, physics, engineering, and computer science. These equations often model real-world phenomena, making their solutions highly valuable.

Understanding Quadratic Equations

Before diving into the specifics, it's essential to understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation are also known as its roots or zeros. These roots represent the values of the variable that satisfy the equation. Solving quadratic equations involves finding these roots, which can be done using various methods, including factoring, completing the square, and the quadratic formula.

Methods for Solving Quadratic Equations

Several methods exist for solving quadratic equations, each with its own advantages and disadvantages. The most common methods include:

Factoring: This method involves expressing the quadratic expression as a product of two linear factors. If the equation can be factored, the roots can be found by setting each factor equal to zero and solving for the variable. Factoring is often the quickest method when it is applicable, but it is not always possible to factor a quadratic equation easily.
Completing the Square: This method involves manipulating the equation algebraically to create a perfect square trinomial on one side. This allows the equation to be written in the form (x + p)^2 = q, where p and q are constants. The roots can then be found by taking the square root of both sides and solving for the variable. Completing the square is a more general method than factoring and can be used to solve any quadratic equation, but it can be more time-consuming.
Quadratic Formula: The quadratic formula is a general formula that provides the solutions to any quadratic equation. The formula is given by x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation. The quadratic formula is the most versatile method for solving quadratic equations, as it can be used for any equation, regardless of whether it can be factored or not. However, it can also be the most computationally intensive method.

Solving t^2 - 100 = 0

Let's begin by solving the equation t^2 - 100 = 0. This equation can be solved using several methods. Here, we'll demonstrate factoring and the square root property.

Factoring Method

The equation t^2 - 100 = 0 can be factored as a difference of squares. Recognizing that 100 is 10 squared, we can rewrite the equation as t^2 - 10^2 = 0. The difference of squares factorization pattern is a^2 - b^2 = (a + b)(a - b). Applying this pattern, we get:

(t + 10)(t - 10) = 0

To find the roots, we set each factor equal to zero:

t + 10 = 0 or t - 10 = 0

Solving for t in each case, we get:

t = -10 or t = 10

Therefore, the solutions to the equation t^2 - 100 = 0 are t = -10 and t = 10.

Square Root Property Method

Another way to solve this equation is by using the square root property. This property states that if x^2 = k, then x = ±√k. We start with the equation:

t^2 - 100 = 0

Add 100 to both sides:

t^2 = 100

Now, take the square root of both sides:

t = ±√100

t = ±10

This gives us the same solutions as the factoring method: t = -10 and t = 10. Both methods effectively solve the equation, demonstrating the flexibility in approaching quadratic equations.

Solving x^2 - 144 = 0

Next, let's tackle the equation x^2 - 144 = 0. Similar to the previous equation, we can use factoring or the square root property to find the solutions.

Factoring Method

Recognizing that 144 is 12 squared, we can rewrite the equation as x^2 - 12^2 = 0. Applying the difference of squares factorization pattern, we get:

(x + 12)(x - 12) = 0

Setting each factor equal to zero:

x + 12 = 0 or x - 12 = 0

Solving for x in each case:

x = -12 or x = 12

Thus, the solutions to the equation x^2 - 144 = 0 are x = -12 and x = 12.

Square Root Property Method

Using the square root property, we start with:

x^2 - 144 = 0

Add 144 to both sides:

x^2 = 144

Take the square root of both sides:

x = ±√144

x = ±12

Again, we find the same solutions: x = -12 and x = 12. This consistency reinforces the reliability of these methods.

Solving t^2 = 81

Now, let's solve the equation t^2 = 81. This equation is already in a form suitable for applying the square root property.

Square Root Property Method

Taking the square root of both sides:

t = ±√81

t = ±9

Therefore, the solutions to the equation t^2 = 81 are t = -9 and t = 9. The simplicity of this equation highlights the power of the square root property.

Solving w^2 = 0

The equation w^2 = 0 is a special case where the square of a variable is equal to zero. This equation has a unique solution.

Square Root Property Method

Taking the square root of both sides:

w = ±√0

w = 0

In this case, there is only one solution: w = 0. This is because zero is the only number whose square is zero.

Solving 5p^2 + 64 = 0

Finally, let's solve the equation 5p^2 + 64 = 0. This equation is slightly more complex as it involves an additional term. We will use algebraic manipulation and the square root property to find the solutions.

Algebraic Manipulation and Square Root Property

First, subtract 64 from both sides:

5p^2 = -64

Next, divide both sides by 5:

p^2 = -64/5

Now, take the square root of both sides:

p = ±√(-64/5)

Since we are taking the square root of a negative number, the solutions will be complex numbers. We can rewrite the expression as:

p = ±√(64/5) * √(-1)

Recall that √(-1) is defined as the imaginary unit, denoted by i. Thus,

p = ±√(64/5) * i

Simplifying the radical, we get:

p = ±(8/√5) * i

To rationalize the denominator, multiply the numerator and denominator by √5:

p = ±(8√5 / 5) * i

Therefore, the solutions to the equation 5p^2 + 64 = 0 are p = (8√5 / 5)i and p = -(8√5 / 5)i. This example demonstrates how to handle equations with complex solutions.

Conclusion

In this article, we have explored various methods for solving quadratic equations and applied them to the equations t^2 - 100 = 0, x^2 - 144 = 0, t^2 = 81, w^2 = 0, and 5p^2 + 64 = 0. We have seen how factoring, the square root property, and algebraic manipulation can be used to find the roots of these equations. Understanding these methods is crucial for solving a wide range of mathematical problems and real-world applications. Mastering these techniques provides a solid foundation for further studies in mathematics and related fields. From simple equations to those with complex solutions, the principles remain the same, offering a consistent approach to problem-solving. The journey through these equations underscores the elegance and power of algebraic methods in unraveling mathematical complexities.