Factor the expression:
$(x + \quad)(x + \quad)$
To factor the quadratic expression \(x^2 + 21x + 54\), we look for an expression of the form \((x + m)(x + n)\). When we expand this product, we obtain:
\[(x + m)(x + n) = x^2 + (m + n)x + mn\]
By comparing this to our given expression \(x^2 + 21x + 54\), we see that we need to find two numbers \(m\) and \(n\) such that:
\[m + n = 21\]
\[mn = 54\]
We look for factors of the constant term, 54, whose sum is equal to the middle coefficient, 21. Let's list the factor pairs of 54:
The pair that satisfies the condition \(m + n = 21\) is \(3\) and \(18\).
With \(m = 3\) and \(n = 18\), we can substitute these values back into the expression \((x + m)(x + n)\) to get:
\[(x + 3)(x + 18)\]
To verify, expand the product:
\[(x + 3)(x + 18) = x^2 + 18x + 3x + 54 = x^2 + 21x + 54\]
This matches our original expression perfectly. The blanks in \((x + \_)(x + \_)\) are \(3\) and \(18\).
Factoring quadratics is a fundamental skill used in physics and engineering to determine critical points. For instance, if you are modeling the height of a projectile, you might have an expression like \(h(t) = -t^2 + 21t + 54\) that represents its altitude over time. Finding when the object hits the ground involves setting this expression to zero and factoring it. Knowing how to decompose these expressions allows you to solve complex real-world motion problems efficiently!