Factor.
$x^2+13 x+42$
$(x-6)(x-7)$
$(x+6)(x-7)$
$(x+7)(x+6)$
$(x+7)(x-6)$
Answer (1)
To factor the quadratic expression x 2 + 13 x + 42 :
Find two numbers that multiply to 42 and add up to 13.
Identify the numbers as 6 and 7 since 6 × 7 = 42 and 6 + 7 = 13 .
Write the factored form as ( x + 6 ) ( x + 7 ) .
The correct factorization is ( x + 7 ) ( x + 6 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 13 x + 42 and asked to factor it. Factoring a quadratic expression means finding two binomials that, when multiplied together, give us the original quadratic expression.
Finding the Right Numbers To factor the quadratic expression x 2 + 13 x + 42 , we need to find two numbers that multiply to 42 (the constant term) and add up to 13 (the coefficient of the x term).
Identifying the Correct Pair Let's list the pairs of factors of 42:
1 and 42 2 and 21 3 and 14 6 and 7
Now, let's check which pair adds up to 13:
1 + 42 = 43 2 + 21 = 23 3 + 14 = 17 6 + 7 = 13
The pair 6 and 7 satisfies both conditions: 6 × 7 = 42 and 6 + 7 = 13 .
Writing the Factored Form Since we found that 6 and 7 are the numbers we need, the factored form of the quadratic expression is ( x + 6 ) ( x + 7 ) . Note that the order of the factors does not matter, so ( x + 7 ) ( x + 6 ) is also correct.
Final Answer Comparing our result with the given options, we see that ( x + 7 ) ( x + 6 ) is one of the options. Therefore, the correct factorization is ( x + 7 ) ( x + 6 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures, ensuring stability and optimal use of materials. Imagine designing a rectangular garden where you know the area is represented by the quadratic expression x 2 + 13 x + 42 . By factoring this expression into ( x + 6 ) ( x + 7 ) , you determine the dimensions of the garden to be ( x + 6 ) and ( x + 7 ) , helping you plan the layout and fencing efficiently. This skill is also crucial in physics for solving projectile motion problems and in economics for modeling supply and demand curves.
