Solving Quadratic Equations: A Comprehensive Guide with Examples

Solving quadratic equations is a fundamental skill in algebra, with applications in various fields such as physics, engineering, and economics. This article will guide you through several methods to solve quadratic equations, including factorization and completing the square. We will also provide examples to illustrate the process.

Introduction to Quadratic Equations

A quadratic equation is any equation that can be written in the form

(ax^2 bx c 0)

where (a), (b), and (c) are constants, and (a eq 0). The solutions to a quadratic equation can be found using various methods, including the quadratic formula, factoring, and completing the square. Here, we will focus on these two methods.

Solving by Factorization

Let's consider the quadratic equation:

(2x^2 - 5x - 3 0)

We can solve this equation by factorization. The goal is to express the quadratic equation as a product of two binomials, i.e.,

((ax b)(cx d) 0)

In this case, we need to find two integers that multiply to (-6) (the product of the coefficient of (x^2) and the constant term) and add to (-5) (the coefficient of (x)). The integers that satisfy these conditions are (-6) and (1).

((2x - 1)(x - 3) 0)

Setting each factor equal to zero gives:

(2x - 1 0)
Solving for (x), we get: (x frac{1}{2}) (x - 3 0)
Solving for (x), we get: (x 3)

The solutions to the quadratic equation are (x frac{1}{2}) and (x 3). To verify these solutions, we can substitute them back into the original equation:

(2left(frac{1}{2}right)^2 - 5left(frac{1}{2}right) - 3 0)

and

(2(3)^2 - 5(3) - 3 0)

Both solutions satisfy the equation, confirming their validity.

Solving by Completing the Square

Another method to solve quadratic equations is completing the square. Let's consider the equation:

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

First, move all terms to one side of the equation:

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

Combine like terms:

(4x^2 - 6x 0)

To complete the square, we first factor out the coefficient of (x^2):

(4x^2 - 6x 0)

Divide every term by 4:

(x^2 - frac{6}{4}x 0)

Find the number to add and subtract inside the parentheses. This number is (left(frac{-6/4}{2}right)^2 left(frac{-3/2}{2}right)^2 frac{9}{16}). Add and subtract this value inside the parentheses:

(x^2 - frac{6}{4}x frac{9}{16} - frac{9}{16} 0)

Write the equation as a perfect square plus a constant:

(left(x - frac{3}{4}right)^2 - frac{9}{16} 0)

Move the constant to the other side:

(left(x - frac{3}{4}right)^2 frac{9}{16})

Take the square root of both sides:

(x - frac{3}{4} pm frac{3}{4})

Finally, solve for (x):

(x frac{3}{4} pm frac{3}{4})

This gives:

(x 0) or (x frac{3}{2})

To verify these solutions, we can substitute them back into the original equation:

(4(0)^2 - 6(0) 0)

and

(4(frac{3}{2})^2 - 6(frac{3}{2}) 0)

Both solutions are correct.

Conclusion

Solving quadratic equations is a critical skill in mathematics, and there are several methods to do so. Factorization and completing the square are two effective techniques. By practicing these methods, you can solve any quadratic equation with confidence. Remember to verify your solutions by substituting them back into the original equation. Happy solving!

Keywords: Solving Quadratic Equations, Factorization, Completing the Square