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To solve quadratic equations like \(x^2 - 4x - 5 = 0\) using the factoring method, the goal is to express the quadratic expression as a product of two binomials. Here's how you do it step-by-step:

1. Identify two numbers that multiply to give you the constant term, \(-5\), and add up to the coefficient of \(x\), which is \(-4\). The two numbers in this case are \(-5\) and \(+1\). 2. Use these numbers to factor the quadratic expression: \((x - 5)(x + 1) = 0\). This is because: - The product \((-5) \cdot 1\) equals \(-5\). - The sum \(-5 + 1\) equals \(-4\).

Once factored, set each binomial equal to zero to find the values of \(x\):

• \(x - 5 = 0\)\(\Rightarrow x = 5\)
• \(x + 1 = 0\)\(\Rightarrow x = -1\)

Thus, the solutions to the equation are \(x = 5\) and \(x = -1\). Factoring is often the simplest method, but it relies on recognizing factorable expressions.

Completing the square is another powerful technique to solve quadratics when factoring is not straightforward. Let's go through the method with our example equation \(x^2 - 4x - 5 = 0\):

1. Rearrange the equation to isolate the linear and quadratic terms on one side. Move the constant term to the other side: \(x^2 - 4x = 5\).
2. Take half of the linear coefficient \(-4\), square it, and add this square to both sides. Here, half of \(-4\) is \(-2\), and its square is \(4\).
3. Add \(4\) to both sides: \(x^2 - 4x + 4 = 5 + 4\).
4. Rewrite the left side as a perfect square trinomial: \((x - 2)^2 = 9\).

By taking the square root of both sides, we simplify the equation to: \(x - 2 = \pm 3\)
Solving for \(x\) gives:

• \(x = 2 + 3 = 5\)
• \(x = 2 - 3 = -1\)

Thus, the solutions are \(x = 5\) and \(x = -1\). Completing the square allows you to handle any quadratic equation methodically.

The quadratic formula is a universal solution for any quadratic equation in the form \(ax^2 + bx + c = 0\). It is particularly useful when the equation is not easily factored. Here's how it works:

The formula itself is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] In our equation, \(x^2 - 4x - 5 = 0\), the values correspond as \(a = 1\), \(b = -4\), and \(c = -5\).

Plug these values into the formula:
1. Calculate the discriminant: \((-4)^2 - 4 \cdot 1 \cdot (-5) = 16 + 20 = 36\). A positive discriminant indicates two real solutions.2. Calculate the roots: - First root: \(x = \frac{4 + 6}{2} = 5\) - Second root: \(x = \frac{4 - 6}{2} = -1\)

The results confirm that the solutions are \(x = 5\) and \(x = -1\). Using the quadratic formula ensures that you find the solutions correctly, no matter how complex the equation might be.