Question
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Tutor-Verified Answer
Answer
\( x^{2} + 14x + 48 = (x + 6)(x + 8) \)
Solution
1. Write the quadratic expression:
\[
x^2 + 14x + 48
\]
2. We look for two numbers whose product is \(48\) and whose sum is \(14\).
3. The numbers \(6\) and \(8\) satisfy these conditions:
\[
6 \times 8 = 48 \quad \text{and} \quad 6 + 8 = 14.
\]
4. Therefore, the factorization is:
\[
x^2 + 14x + 48 = (x + 6)(x + 8).
\]
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Simplify this solution The Deep Dive
To factor the quadratic expression \( x^{2} + 14x + 48 \), we need to find two numbers that multiply to 48 (the constant term) and add up to 14 (the coefficient of \( x \)). The numbers 6 and 8 fit the bill since \( 6 \times 8 = 48 \) and \( 6 + 8 = 14 \). Therefore, the factored form is \( (x + 6)(x + 8) \). These factors can help find the roots of the quadratic equation \( x^2 + 14x + 48 = 0 \) more easily—just set each factor to zero, giving you \( x + 6 = 0 \) or \( x + 8 = 0 \). This leads to the solutions \( x = -6 \) and \( x = -8 \). Factoring not only simplifies the expression but also provides valuable insights into its roots!
