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Factoring a quadratic equation is a method to find the solutions of the equation by expressing it as the product of its factors. The standard form of a quadratic equation is given by \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients. To factor the quadratic equation \(x^2 - 5x - 6 = 0\), we look for two numbers whose product is \(c = -6\) and whose sum is \(b = -5\).

• These numbers are \(-6\) and \(1\).
• They satisfy the condition \(p \times q = -6\) and \(p + q = -5\).

Thus, the equation can be rewritten as \(x^2 - 5x - 6 = (x - 6)(x + 1) = 0\).
The factors \(x - 6\) and \(x + 1\) help us solve for the roots of the equation by setting each factor equal to zero. This results in two solutions for \(x\): \(x = 6\) and \(x = -1\).
Factoring is a powerful and efficient method to solve quadratic equations, especially when the numbers are easy to recognize and factor.

Once we have found the possible solutions \(x = 6\) and \(x = -1\) by factoring, it is crucial to verify them. Verification ensures that both solutions truly satisfy the original quadratic equation. We do this by substituting each solution back into the equation. Let's examine both solutions:

• Substitute \(x = 6\):
  \[6^2 - 5(6) - 6 = 36 - 30 - 6 = 0\]
• Substitute \(x = -1\):
  \[(-1)^2 - 5(-1) - 6 = 1 + 5 - 6 = 0\]

In both cases, the results are \(0\), confirming that both \(x = 6\) and \(x = -1\) are valid solutions of the equation. Verification of solutions is an essential step in solving quadratic equations to ensure accuracy and correctness.

Understanding the standard form of a quadratic equation is fundamental in solving and manipulating these types of equations. The standard form is written as \(ax^2 + bx + c = 0\), where:

• \(a\) is the coefficient of \(x^2\) and determines the direction and width of the parabola.
• \(b\) is the coefficient of \(x\), influencing the axis of symmetry and the vertex.
• \(c\) is the constant term, indicating the equation's value when \(x = 0\).

In the problem \(x^2 - 5x - 6 = 0\), the coefficients are \(a = 1\), \(b = -5\), and \(c = -6\). Recognizing and writing a quadratic equation in its standard form makes it easier to apply methods like factoring and using the quadratic formula. It allows us to quickly identify the roots of the equation by solving for the zeros of the function. The standard form serves as the starting point for any further analysis or calculation related to quadratic equations.