Solving the Quadratic Equation x^2 - 4x - 12 0: Methods and Applications

In this article, we delve into the solutions of the quadratic equation x^2 - 4x - 12 0 through various methods, explaining each step in detail. Whether you are a beginner in algebra or looking to refresh your understanding, this guide will provide clear and comprehensive insights.

1. Solving Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form ax^2 bx c 0. For our equation x^2 - 4x - 12 0, we identify the coefficients as:

a 1 b -4 c -12

Step 1: Calculate the Discriminant

The discriminant Δ b^2 - 4ac helps determine the nature of the roots:

Δ (-4)^2 - 4(1)(-12) Δ 16 48 64

The discriminant being positive indicates that the equation has two distinct real roots.

Step 2: Apply the Quadratic Formula

The quadratic formula is given by:

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

Substituting the values, we get:

x frac{-(-4) pm sqrt{64}}{2(1)}

This simplifies to:

x frac{4 pm 8}{2}

Therefore, the two solutions are:

x frac{4 8}{2} 6 x frac{4 - 8}{2} -2

2. Factoring the Quadratic Equation

Another method to solve x^2 - 4x - 12 0 is through factorization. This involves expressing the quadratic equation as a product of two binomials.

Given the equation:

x^2 - 4x - 12 0

We need to find two numbers that multiply to -12 and add to -4. These numbers are -6 and 2.

Thus, we can factor the equation as:

(x - 6)(x 2) 0

Setting each factor to zero gives:

x - 6 0 Rightarrow x 6 x 2 0 Rightarrow x -2

3. Completing the Square

Completing the square is another approach that transforms the quadratic equation into a perfect square trinomial.

Starting with the equation:

x^2 - 4x - 12 0

We first isolate the constant term:

x^2 - 4x 12

To complete the square, we add and subtract the square of half the coefficient of x (which is 2), resulting in:

x^2 - 4x 4 - 4 12

This simplifies to:

(x - 2)^2 - 4 12

Adding 4 to both sides:

(x - 2)^2 16

Taking the square root of both sides:

x - 2 pm 4

Solving for x gives:

x - 2 4 Rightarrow x 6 x - 2 -4 Rightarrow x -2

Conclusion

By using the quadratic formula, factoring, and completing the square, we have found that the solutions to the equation x^2 - 4x - 12 0 are x 6 and x -2. Understanding these methods can greatly enhance your algebra skills and problem-solving abilities.

Should you have any doubts or require further assistance, don't hesitate to seek help from your peers or a math tutor. Happy solving!