Verified Answer :
- Expand the expression \((x+p)(x+q)\) to get \(x^2 + (p+q)x + pq\).
- Compare this expansion to the given expression \(x^2 + bx + c\).
- Match the coefficients to identify that \(p+q = b\) and \(pq = c\).
- The required values are \(\boxed{pq = c, p+q = b}\).
Explanation
- Analyzing the Problem
We are given a quadratic expression of the form \(x^2 + bx + c\) and wish to express it in its factored form \((x+p)(x+q)\). This task involves finding the relationships between the constants \(b\) and \(c\) in the original expression and the parameters \(p\) and \(q\) in the factored form.
- Expanding the Factored Form
To find the relationship, we expand the factored form \((x+p)(x+q)\) using the distributive property, often referred to as the FOIL method:
\((x+p)(x+q) = x(x) + x(q) + p(x) + p(q)\)
\((x+p)(x+q) = x^2 + xq + px + pq\)
Combining the terms that contain \(x\), we get:
\((x+p)(x+q) = x^2 + (p+q)x + pq\)
- Comparing Coefficients
Now, we compare our expanded expression \(x^2 + (p+q)x + pq\) with the original expression \(x^2 + bx + c\). For these two expressions to be identical for all values of \(x\), their corresponding coefficients must be equal:
- The coefficient of \(x\) must match: \(p + q = b\)
- The constant term must match: \(pq = c\)
Thus, for the factorization \(x^2 + bx + c = (x+p)(x+q)\) to hold true, we must have \(p+q = b\) and \(pq = c\).
- Conclusion and Final Answer
In conclusion, the factors \((x+p)\) and \((x+q)\) are determined by finding two numbers \(p\) and \(q\) such that their sum equals the coefficient of the linear term (\(b\)) and their product equals the constant term (\(c\)). Therefore, the blanks are \(pq = c\) and \(p+q = b\). The final answer is \(pq = c\) and \(p+q = b\).
Examples
Factoring quadratics is a foundational skill in algebra that allows us to solve complex equations and simplify expressions. For example, if you are designing a rectangular garden with an area given by \(x^2 + 5x + 6\), factoring it into \((x+2)(x+3)\) helps you determine the dimensions of the garden as \((x+2)\) and \((x+3)\). This allows you to easily find the dimensions for any given length \(x\). This process is widely used in physics, engineering, and economics to model relationships and solve for unknown variables efficiently.