Factor.
$x^2-13 x+42$
$(x-6)(x-7)$
$(x+6)(x+7)$
$(x-7)(x+6)$
$(x-6)(x+7)$
Answer (1)
Find two numbers that multiply to 42 and add up to -13.
The two numbers are -6 and -7.
Write the factored form using these numbers: ( x − 6 ) ( x − 7 ) .
The factorization of the quadratic expression is ( x − 6 ) ( x − 7 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 − 13 x + 42 and asked to factor it. Factoring a quadratic expression involves finding two binomials that, when multiplied together, give us the original quadratic expression.
Finding the Correct Factors We need to find two numbers that multiply to 42 (the constant term) and add up to -13 (the coefficient of the x term). Let's list the factor pairs of 42:
1 and 42 2 and 21 3 and 14 6 and 7
Since we need the two numbers to add up to a negative number (-13), both numbers must be negative. So, let's consider the negative factor pairs:
-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
So, the two numbers are -6 and -7.
Writing the Factored Form Now that we have the two numbers, -6 and -7, we can write the factored form of the quadratic expression as ( x − 6 ) ( x − 7 ) .
Final Answer Therefore, the factorization of x 2 − 13 x + 42 is ( x − 6 ) ( x − 7 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, engineers use factoring to design structures and predict their behavior under different loads. Imagine you are designing a rectangular garden with an area represented by the quadratic expression x 2 − 13 x + 42 . By factoring this expression into ( x − 6 ) ( x − 7 ) , you determine the possible dimensions of the garden. If x represents a certain length, then ( x − 6 ) and ( x − 7 ) would be the width and length of the garden, respectively. This allows you to plan the layout and optimize the use of space.
