What is the correct factorization of $x^2+5 x-6$ ?
To factorize the quadratic expression \(x^2 + 5x - 6\), we look for two numbers, \(p\) and \(q\), that satisfy the conditions required by the general form of a quadratic expression \(x^2 + bx + c = (x + p)(x + q)\). In this form, the sum of the numbers must be equal to the coefficient of the middle term (\(b = 5\)), and their product must be equal to the constant term (\(c = -6\)). Therefore, we need to find two numbers such that:
\[p + q = 5\]
\[p \cdot q = -6\]
We examine the factors of \(-6\) and check their sums to see which pair satisfies the requirement \(p + q = 5\).
The pair \(-1\) and \(6\) adds up to \(5\), which perfectly matches the middle coefficient \(b = 5\). Since their product is \((-1) \times 6 = -6\), these are the correct numbers to use for the factorization.
Having identified \(p = -1\) and \(q = 6\), we can write the factored form of the expression as:
\[x^2 + 5x - 6 = (x - 1)(x + 6)\]
By expanding this, we get \(x \cdot x + x \cdot 6 - 1 \cdot x - 1 \cdot 6 = x^2 + 6x - x - 6 = x^2 + 5x - 6\), which confirms our result. Comparing this with the provided options:
The correct factorization corresponds to option (d).
Factoring quadratic expressions is a fundamental skill in algebra with many real-world applications. For example, if you are designing a rectangular garden and know that the area can be represented by the expression \(x^2 + 5x - 6\), factoring it helps you determine the dimensions of the garden as \((x-1)\) and \((x+6)\). This allows you to understand how changing the value of \(x\) affects the length and width of your design, which is essential in fields like landscape architecture, engineering, and resource planning.