Factor.
$x^2+12 x+35$
A. $(x+7)(x-5)$
B. $(x-7)(x-5)$
C. $(x+5)(x-7)$
D. $(x+5)(x+7)$
Answer (1)
To factor the quadratic expression x 2 + 12 x + 35 , we need to find two numbers that multiply to 35 and add to 12.
The numbers are 5 and 7.
Therefore, the factored form is ( x + 5 ) ( x + 7 ) .
Expanding ( x + 5 ) ( x + 7 ) confirms that it equals x 2 + 12 x + 35 .
The final answer is ( x + 5 ) ( x + 7 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 12 x + 35 and asked to factor it. Factoring a quadratic means expressing it as a product of two binomials.
Finding the Correct Numbers We need to find two numbers that multiply to 35 (the constant term) and add up to 12 (the coefficient of the x term). Let's list the factor pairs of 35:
1 and 35 5 and 7
Writing the Factored Form The pair of factors that add up to 12 are 5 and 7, since 5 + 7 = 12 . Therefore, the factored form of the quadratic expression is ( x + 5 ) ( x + 7 ) .
Checking the Answer We can check our answer by expanding the factored form:
( x + 5 ) ( x + 7 ) = x ( x + 7 ) + 5 ( x + 7 ) = x 2 + 7 x + 5 x + 35 = x 2 + 12 x + 35
This matches the original quadratic expression, so our factorization is correct.
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has 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 and want to determine the possible dimensions. Factoring the quadratic expression representing the area can help you find the possible lengths and widths of the garden, making your design process more efficient and precise. This skill is also crucial in physics for solving problems related to projectile motion and other scenarios involving quadratic relationships.
