Factor the quadratic $x^2+2 x-15$. List ONLY ONE of the factors with parentheses and no spaces. Example: $(x+1)$
Answer (2)
Find two numbers that add up to 2 and multiply to -15. These numbers are 5 and -3.
Write the factored form of the quadratic using these numbers: ( x + 5 ) ( x − 3 ) .
List one of the factors: ( x + 5 ) .
The final answer is ( x + 5 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 2 x − 15 . Our goal is to factor this quadratic expression into two binomial factors.
Finding the Numbers We need to find two numbers that add up to 2 (the coefficient of the x term) and multiply to -15 (the constant term). Let's call these numbers a and b . So, we need to find a and b such that:
a + b = 2 a × b = − 15
Identifying the Correct Pair By trying different pairs of numbers, we find that 5 and -3 satisfy these conditions:
5 + ( − 3 ) = 2 5 × ( − 3 ) = − 15
Writing the Factored Form Now that we have the numbers 5 and -3, we can write the factored form of the quadratic expression as:
( x + 5 ) ( x − 3 )
Listing One Factor The problem asks us to list only one of the factors. We can choose either ( x + 5 ) or ( x − 3 ) . Let's choose ( x + 5 ) .
Examples
Factoring quadratics is useful in many real-world scenarios, such as determining the dimensions of a garden given its area. For example, if you know the area of a rectangular garden is represented by the quadratic x 2 + 2 x − 15 , factoring it into ( x + 5 ) ( x − 3 ) helps you find possible lengths and widths of the garden. If x represents a length, then one side could be x + 5 and the other x − 3 . This type of problem also appears in physics when analyzing projectile motion, where factoring can help determine when an object hits the ground.
The quadratic x 2 + 2 x − 15 factors into ( x + 5 ) ( x − 3 ) . We can list one of the factors, which is ( x + 5 ) .
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