Solving Quadratic Equations with Variable Roots: A Comprehensive Guide

Quadratic equations are a fundamental part of algebra, and understanding how to solve them is crucial for students and professionals alike. This article explores various methods to solve quadratic equations, focusing on the importance of considering all possible roots.

Introduction to Quadratic Equations

A quadratic equation is an equation of the form (ax^2 bx c 0), where (a), (b), and (c) are constants, and (a eq 0). The solutions to these equations are the roots, or values of (x) that satisfy the equation.

Example 1: Solving (x^{2x} - 1 frac{1}{2}^{2} - 1 - frac{1}{4})

Given the equation (x^{2x} - 1 frac{1}{2}^{2} - 1 - frac{1}{4}), we can simplify it as follows:

1. Compute (frac{1}{2}^{2} - 1 - frac{1}{4}):

(frac{1}{2}^{2} frac{1}{4})

(frac{1}{4} - 1 - frac{1}{4} -1)

Solution Steps:

1. Therefore, the equation simplifies to:

(x^{2x} - 1 -1)

2. Simplify further:

(x^{2x} 0)

This equation does not have a real solution because any non-zero number raised to any power is never zero.

Example 2: Solving (x - sqrt{x^2} x)

To solve the equation (x - sqrt{x^2} x), follow these steps:

1. Recall that (sqrt{x^2} |x|):

2. The equation becomes:

(x - |x| x)

3. Simplify the left side:

If (x geq 0), then (|x| x), and the equation simplifies to:

(x - x x) (Rightarrow 0 x)

This solution is not valid because it contradicts (x - |x| x).

If (x

(x - (-x) x) (Rightarrow 2x x)

(x 0)

Therefore, the only valid solution is (x 0).

Example 3: Solving (x^2 - 4 1)

To solve the equation (x^2 - 4 1), follow these steps:

1. Isolate (x^2):

(x^2 5)

2. Take the square root of both sides:

(x pm sqrt{5})

Therefore, the solutions are:

(x frac{-1 sqrt{5}}{2}) and (x frac{-1 - sqrt{5}}{2})

Example 4: Solving (x^2 - 4x 4 4)

To solve the equation (x^2 - 4x 4 4), follow these steps:

1. Simplify the equation:

(x^2 - 4x 4 - 4 0)

(x^2 - 4x 0)

2. Factor the equation:

(x(x - 4) 0)

3. Solve for (x):

(x 0) or (x 4)

Example 5: Solving (x^2 - 1 2)

To solve the equation (x^2 - 1 2), follow these steps:

1. Isolate (x^2):

(x^2 3)

2. Take the square root of both sides:

(x pm sqrt{3})