91Ó°ÊÓ

Factoring is a popular method used to solve quadratic equations by expressing them as a product of their linear factors. For the quadratic equation \(x^{2} - 4x - 5 = 0\), the first step is to identify two numbers that multiply to \ -5 (the constant term) and add up to \ -4 (the coefficient of the linear term x). These numbers are \(-5\) and \(1\).
Next, rewrite the quadratic equation as \ \((x - 5)(x + 1) = 0\), indicating that it has been factored into two binomials. \ These binomials, \((x - 5)\) and \((x + 1)\), set the stage for the next step.
To find the solutions for \(x\), set each factor equal to zero:

• \(x - 5 = 0\): Solving this gives \(x = 5\).
• \(x + 1 = 0\): Solving this gives \(x = -1\).

The factorization method confirms the solutions \(x = 5\) and \(x = -1\). Factoring is efficient but not always applicable, particularly for non-factorable quadratics. However, it's quite handy when it works, especially with neat integers in play.

Completing the square transforms a quadratic equation into a perfect square trinomial, which is much simpler to solve. Starting with the equation \(x^{2} - 4x - 5 = 0\), the goal is to rearrange the terms into a perfect square form.
Firstly, shift the constant term to the right side by moving -5: \ \(x^{2} - 4x = 5\).
To complete the square, calculate the term to add to both sides, using the formula \ \((\frac{b}{2})^2\). Here, \ \(b = -4\), so \ \((\frac{-4}{2})^2 = 4\).
Add 4 to both sides, delivering the equation \ \(x^{2} - 4x + 4 = 9\).
This allows us to write the equation as a squared binomial: \ \((x - 2)^{2} = 9\).
Finally, solve for \(x\) by taking the square root of both sides:

• \(x - 2 = 3\), which resolves to \(x = 5\).
• \(x - 2 = -3\), which resolves to \(x = -1\).

Thus, the same solutions \(x = 5\) and \(x = -1\) are achieved. Completing the square clearly demonstrated the transformation and solution of these quadratic equations, being particularly useful in equations where the quadratic coefficient isn’t one.

The quadratic formula provides a universal method to solve any quadratic equation, \(ax^{2} + bx + c = 0\). It's given by the formula:
\[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]For the given equation \(x^{2} - 4x - 5 = 0\), we identify \ \(a = 1\), \ \(b = -4\), and \ \(c = -5\).

The next step involves calculating the discriminant, \ \(b^{2} - 4ac\), crucial for determining the nature and number of solutions:

• \((-4)^{2} - 4(1)(-5) = 16 + 20 = 36\)

Since the discriminant is positive, there are two real and distinct solutions.
Substitute these values into the quadratic formula:

• \(x = \frac{-(-4) \pm \sqrt{36}}{2 \times 1} = \frac{4 \pm 6}{2}\)

Solving these gives the solutions:

• \(x = \frac{4 + 6}{2} = 5\)
• \(x = \frac{4 - 6}{2} = -1\)

The quadratic formula is highly valuable as it guarantees a solution for any quadratic equation, making it indispensable when other methods, like factoring, are not feasible or straightforward.