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A quadratic expression is a polynomial expression where the highest power of the variable is 2. It typically follows the form \( ax^2 + bx + c \). In this case, we're focusing on expressions of this form with a being non-zero, which ensures it's truly quadratic.
Quadratic expressions are fundamental in algebra and often arise in various real-world problems. Recognizing the quadratic form makes it easier to apply formulas or solve equations related to it.
In the given problem, we have the quadratic expression \( u^2 - 5u - 6 \). Here, the coefficients are \( a = 1 \), \( b = -5 \), and \( c = -6 \). These numbers help determine how the parabola, represented by the quadratic, opens and where its vertex lies.

A binomial is a polynomial expression with exactly two terms. When factoring quadratic expressions like \( u^2 - 5u - 6 \), we aim to rewrite the expression as a product of two binomials.
Each binomial typically takes the form \( (u + p)(u + q) \), where \( p \) and \( q \) are constants. The challenge is finding the right combination of \( p \) and \( q \) such that their product is equal to \( c \) and their sum is equal to \( b \). This involves a bit of trial and error or a keen eye for numbers.
In this exercise, the numbers \( -6 \) and \( 1 \) were chosen because \((-6) \times 1 = -6 \) and \((-6) + 1 = -5 \). Thus, the expression can be rewritten as \( (u - 6)(u + 1) \). This binomial pair, when multiplied, returns the original quadratic equation.

The factored form of a quadratic expression is when it is expressed as the product of two binomials. This form is particularly useful because it reveals the roots or solutions to the quadratic equation easily.
For the expression \( u^2 - 5u - 6 \), the factored form is \( (u-6)(u+1) \). By setting each binomial to zero, we can find the values of \( u \) that satisfy the equation \( u^2 - 5u - 6 = 0 \). These values \( u = 6 \) and \( u = -1 \) are the points where the parabola crosses the u-axis in the coordinate plane.
Factoring reduces the complexity of the equation and offers a direct path to finding solutions, helping in graphing or solving algebraic equations.