Solving Quadratic Equations Using the Factoring Method

When solving quadratic equations, the factoring method is a powerful tool. A quadratic equation is of the form ax^2 bx c 0, where a, b, and c are constants, and a ≠ 0. This article will walk you through solving various quadratic equations using the factoring method, providing a clear understanding of the process and how to apply it effectively.

Understanding the Factoring Method

The factoring method involves breaking down the quadratic equation into simpler factors, which can then be set to zero to solve for the unknown variable. The key is to find two numbers that multiply to the constant term (c) and add up to the coefficient of the linear term (b).

Solving Specific Quadratic Equations

1. x^2 - 11 0

This equation requires a bit of manipulation to make it factorable:

[x^2 - 11 0 Rightarrow x^2 - 11x - 11 0 Rightarrow x(x - 11) 0]

Setting each factor to zero gives the solutions:

[x 0 quad text{or} quad x 11]

2. x^2 - 2x - 24 0

For this equation, we need to find two numbers that multiply to -24 and add to -2. These numbers are -6 and 4:

[x^2 - 2x - 24 0 Rightarrow (x - 6)(x 4) 0]

Setting each factor to zero gives:

[x 6 quad text{or} quad x -4]

3. x^2 3x - 18 0

For this equation, we need numbers that multiply to -18 and add to 3. These numbers are 6 and -3:

[x^2 3x - 18 0 Rightarrow (x 6)(x - 3) 0]

Setting each factor to zero gives:

[x -6 quad text{or} quad x 3]

4. x^2 - 11x - 18 0

The numbers that multiply to -18 and add to -11 are 9 and 2:

[x^2 - 11x - 18 0 Rightarrow (x - 9)(x 2) 0]

Setting each factor to zero gives:

[x 9 quad text{or} quad x -2]

5. x^2 - 8x - 15 0

For this equation, the numbers that multiply to -15 and add to -8 are 5 and 3:

[x^2 - 8x - 15 0 Rightarrow (x - 5)(x 3) 0]

Setting each factor to zero gives:

[x 5 quad text{or} quad x -3]

Conclusion

The factoring method is a straightforward and effective way to solve quadratic equations. By breaking down the quadratic into simpler factors, we can easily find the roots of the equation. The key is to identify the correct numbers that multiply to the constant term and add to the coefficient of the linear term. This method, combined with clear steps, can be a valuable asset in solving various quadratic equations.

Further Reading

For more detailed information and practice problems, refer to resources such as Khan Academy, Math Is Fun, and Purplemath. These websites offer comprehensive guides and exercises to help you master the factoring method and other algebraic techniques.