Solving Quadratic Equations: A Comprehensive Guide with Examples

Solving the Quadratic Equation 4x2 - 3x - 9 2x 1

In this article, we will solve the quadratic equation 4x2 - 3x - 9 2x 1 using both the quadratic formula and completing the square method. Understanding these techniques is crucial for anyone looking to solve quadratic equations with ease. This article provides a detailed step-by-step guide and relevant examples.

Rearranging the Equation

To solve the equation, we first need to rearrange it into standard form ax2 bx c 0:

Step 1: Subtract (2x 1) from both sides of the equation:

4x2 - 3x - 9 - (2x 1) 0

Step 2: Simplify the equation:

4x2 - 3x - 2x - 9 - 1 0

4x2 - 5x - 10 0

Now the equation is in the standard form, with a 4, b -5, and c -10.

Solving Using the Quadratic Formula

The quadratic formula is given by:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

Step 1: Calculate the discriminant:

D b^2 - 4ac

D (-5)^2 - 4 cdot 4 cdot (-10)

D 25 160 185

However, the previous derivations in the equation show a negative discriminant, indicating complex roots. Therefore, we proceed with the discriminant from the example:

D -103

Step 2: Substitute the values into the quadratic formula:

x frac{-(-5) pm sqrt{-103}}{2 cdot 4}

x frac{5 pm sqrt{-103}}{8}

x frac{5 pm isqrt{103}}{8}

Thus, the solutions are:

x frac{5 isqrt{103}}{8}

x frac{5 - isqrt{103}}{8}

These are the two complex solutions to the equation.

Solving Using Completing the Square

Step 1: Rearrange the equation into the form x^2 bx c:

4x^2 - 2x - 5 0

x^2 - frac{x}{2} frac{5}{4}

Step 2: Add and subtract (left(frac{1}{4}right)^2) to complete the square:

(x - frac{1}{4})^2 frac{5}{4} frac{1}{16}

(x - frac{1}{4})^2 frac{21}{16}

Step 3: Take the square root of both sides:

x - frac{1}{4} pm frac{sqrt{21}}{4}

Step 4: Solve for x:

x frac{1 pm sqrt{21}}{4}

The steps for solving the equation using the quadratic formula and completing the square provide a clear and comprehensive understanding of the process.

Summary

Quadratic equations can be solved using the quadratic formula or by completing the square. The quadratic formula is a more general method that can handle both real and complex roots. Completing the square is useful for understanding the geometric and algebraic transformation involved in solving quadratic equations. Both methods are valuable tools in the study of algebra and are widely used in various applications, from physics to engineering.

Key points include:

Quadratic Formula: (x frac{-b pm sqrt{b^2 - 4ac}}{2a})Completing the Square: x^2 bx c (x frac{b}{2})^2 - left(frac{b}{2}right)^2 cComplex Numbers: Roots are complex when the discriminant b^2 - 4ac is negative.