Factor the quadratic $x^2+1x-12$. List ONLY ONE of the factors with parentheses and no spaces. Example: $(x+1)$
Answer (1)
We need to factor the quadratic expression x 2 + x − 12 .
We look for two numbers that multiply to -12 and add to 1. These numbers are -3 and 4.
The factors are ( x − 3 ) and ( x + 4 ) .
One of the factors is ( x + 4 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression x 2 + x − 12 . This means we want to find two binomials of the form ( x + a ) and ( x + b ) such that when we multiply them together, we get x 2 + x − 12 .
Finding the Numbers To find these two binomials, we need to find two numbers, a and b , that satisfy two conditions:
Their product is equal to the constant term of the quadratic, which is -12.
Their sum is equal to the coefficient of the x term, which is 1.
Identifying the Correct Pair Let's list pairs of factors of -12:
1 and -12 (sum is -11)
-1 and 12 (sum is 11)
2 and -6 (sum is -4)
-2 and 6 (sum is 4)
3 and -4 (sum is -1)
-3 and 4 (sum is 1)
We see that the pair -3 and 4 satisfy both conditions: ( − 3 ) × 4 = − 12 and ( − 3 ) + 4 = 1 .
Writing the Factors Therefore, the factors of the quadratic expression are ( x − 3 ) and ( x + 4 ) .
Listing One Factor The problem asks us to list only one of the factors. We can choose either ( x − 3 ) or ( x + 4 ) .
Examples
Factoring quadratics is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and predict their behavior under different loads. Architects use factoring to create aesthetically pleasing and structurally sound buildings. Financial analysts use factoring to model and predict market trends. By mastering factoring, you'll be equipped to solve a wide range of problems in various fields.
