Fill in the gaps to factorise this expression.
[tex]x^2+4 x-21=(x-\ldots)(x+\ldots)[/tex]
Answer (1)
Find two numbers that multiply to -21 and add to 4. These numbers are -3 and 7.
Write the factored form using these numbers: ( x − 3 ) ( x + 7 ) .
Verify the factorization by expanding: ( x − 3 ) ( x + 7 ) = x 2 + 4 x − 21 .
The factored expression is ( x − 3 ) ( x + 7 ) , so the missing numbers are 3 and 7 .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 4 x − 21 and asked to factor it into the form ( x − a ) ( x + b ) , where a and b are numbers to be determined. Our goal is to find the values of a and b that correctly factorize the expression.
Finding the Numbers To factor the quadratic expression x 2 + 4 x − 21 , we need to find two numbers that multiply to -21 (the constant term) and add up to 4 (the coefficient of the x term).
Determining the Correct Pair Let's list the factor pairs of 21: 1 and 21, 3 and 7. Since the product needs to be -21, one of the numbers must be negative. We need to find a pair where the difference between the numbers is 4. The pair 3 and 7 works, since 7 − 3 = 4 . Thus, we can use -3 and 7 as our numbers, since − 3 × 7 = − 21 and − 3 + 7 = 4 .
Writing the Factorization Now we can write the factorization as ( x − 3 ) ( x + 7 ) . Let's check our work by expanding this expression: ( x − 3 ) ( x + 7 ) = x 2 + 7 x − 3 x − 21 = x 2 + 4 x − 21 This matches the original expression, so our factorization is correct.
Final Answer Therefore, the factorization of x 2 + 4 x − 21 is ( x − 3 ) ( x + 7 ) . The missing numbers are 3 and 7.
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has 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. Imagine you are designing a rectangular garden with an area represented by the expression x 2 + 4 x − 21 . By factoring this expression into ( x − 3 ) ( x + 7 ) , you determine the dimensions of the garden, ensuring it fits perfectly in your yard. This skill is also crucial in physics for solving projectile motion problems, where factoring helps determine the time it takes for an object to hit the ground.
