Factor the quadratic $x^2-3 x-10$. List ONLY ONE of the factors with parentheses and no spaces. Example: $(x+1)$
Answer (2)
We need to factor the quadratic expression x 2 − 3 x − 10 .
Find two numbers that multiply to -10 and add to -3. These numbers are 2 and -5.
Write the quadratic in factored form: ( x + 2 ) ( x − 5 ) .
List one of the factors: ( x + 2 )
Explanation
Understanding the Problem We are given the quadratic expression x 2 − 3 x − 10 . Our goal is to factor this quadratic expression into two binomials. We need to find two numbers that multiply to -10 and add up to -3.
Finding the Numbers Let's find two numbers whose product is -10 and whose sum is -3. We can list the factor pairs of -10:
1 and -10 (sum is -9)
-1 and 10 (sum is 9)
2 and -5 (sum is -3)
-2 and 5 (sum is 3)
The pair of numbers that satisfy our condition are 2 and -5.
Factoring the Quadratic Now we can write the quadratic expression in factored form using the numbers we found: x 2 − 3 x − 10 = ( x + 2 ) ( x − 5 ) We can check our factoring by expanding the expression: ( x + 2 ) ( x − 5 ) = x 2 − 5 x + 2 x − 10 = x 2 − 3 x − 10 This matches the original quadratic expression, so our factoring is correct.
Listing One Factor The problem asks us to list only one of the factors. We can choose either ( x + 2 ) or ( x − 5 ) .
Examples
Factoring quadratics is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to design efficient algorithms. Factoring also helps in solving projectile motion problems, where you need to find when an object hits the ground, which involves solving a quadratic equation.
To factor the quadratic expression x 2 − 3 x − 10 , we find two numbers, 2 and − 5 , that multiply to − 10 and add to − 3 . This allows us to express the quadratic as ( x + 2 ) ( x − 5 ) . One of the factors is (\boxed{(x + 2)}.
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