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Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The variable \( x \) represents an unknown value that we want to solve for. In this type of equation, the highest power of \( x \) is 2, which is what makes the equation a "quadratic." This form allows for a wide range of solutions, depending on the coefficients.

• Understanding the coefficients: \( a \) represents the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term.
• Solving methods: You can solve quadratic equations by factoring, using the quadratic formula, completing the square, or graphing.

In our given problem, the quadratic equation is \( x^2 - 11x + 28 = 0 \). Recognizing the equation as a quadratic is the first step to solving it by methods like factoring.

When we factor a trinomial, we want to express it as a product of binomials. A binomial is a polynomial with two terms, such as \((x - 4)\) or \((x - 7)\). The objective of factorization is to break down a complex expression into simpler multiplicative components that can be handled more easily.To factor the trinomial \( x^2 - 11x + 28 \) into binomials, we need two numbers that multiply to 28 (the constant term) and add up to -11 (the coefficient of \( x \)). In this case, the numbers -4 and -7 fulfill these requirements:

• Multiplication check: \((-4) \times (-7) = 28\)
• Addition check: \((-4) + (-7) = -11\)

Therefore, we rewrite the trinomial as \((x - 4)(x - 7)\), which are two binomial factors. Breaking the trinomial into these factors simplifies certain operations, such as graphing or solving for roots.

Trinomial factorization involves breaking down a three-term polynomial (of the form \( ax^2 + bx + c \)) into simpler expressions. This process is essential in algebra as it simplifies equations and aids in solving them by finding roots.For the trinomial \( x^2 - 11x + 28 \), factorization is straightforward because \( a = 1 \). Here’s a quick recap of how we achieved the factorization:

• Identify suitable numbers that multiply to \( ac = 28 \) and sum to \( b = -11 \).
• These calculations lead us to -4 and -7.

We used these values to express the trinomial as a product of two binomials: \((x - 4)(x - 7)\). This factorization ensures that any further operations—like solving for \( x \)—look at each binomial independently, offering solutions faster. Verify factorization by expanding the factors to double-check your work.