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Factoring is a crucial technique in solving quadratic equations. It involves breaking down a complex expression into simpler parts that when multiplied together give back the original expression. In the context of quadratic equations, it helps us identify roots or zeros of the equation by setting the factors equal to zero.
To factor the equation, ensure that it is in the standard form, which means having it look like this: \(ax^2 + bx + c = 0\). For example, if you have the equation \(x^2 - 5x = 84\), rearrange it to \(x^2 - 5x - 84 = 0\).
The next step is to find pairs of numbers that multiply to the constant term \(c\) (here, \(-84\)) and add up to the linear coefficient \(b\) (here, \(-5\)). In this example, \(-12\) and \(7\) are the numbers that satisfy these conditions:

• \(-12 \times 7 = -84\)
• \(-12 + 7 = -5\)

Therefore, the factored form of the equation \(x^2 - 5x - 84\) is \((x - 12)(x + 7) = 0\).

Quadratic equations are a specific type of polynomial equations that involve a variable raised to the power of two. They take the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients where \(a eq 0\).
This characteristic makes quadratic equations appear as parabolas when graphed on a Cartesian plane. Solving these equations often involves factoring, using the quadratic formula, or completing the square.
The importance of polynomial equations extends beyond quadratic equations. They form the basis for more complex polynomials which can describe a wide range of mathematical phenomena. Understanding how to manipulate and solve them is key to a deep understanding of algebra.

Algebraic solutions refer to solving equations by finding the values of the unknown variables that make the equation true. For quadratic equations, once you factor it successfully as \((x - 12)(x + 7) = 0\), you proceed by solving each factor for the variable \(x\).
This involves setting each individual factor equal to zero and solving:

• \(x - 12 = 0\)
• \(x + 7 = 0\)

Solving these gives you \(x = 12\) and \(x = -7\) respectively. These are the algebraic solutions to the quadratic equation \(x^2 - 5x = 84\).
Algebraic solutions provide concrete values which satisfy the original equation, demonstrating how algebra can be used to solve real-world problems through precise calculations. Knowing different methods to arrive at these solutions is fundamental for solving more complex problems.