One of the steps needed to solve a quadratic equation by grouping is to split the middle term as a sum of two terms. Given the equation x2−13x+42=0%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A13%0D%0A𝑥%0D%0A+%0D%0A42%0D%0A=%0D%0A0%0D%0A , which two numbers would be needed to complete this step? Enter the lesser value first.(1 point)%0D%0Ax2−13x+42=0%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A13%0D%0A𝑥%0D%0A+%0D%0A42%0D%0A=%0D%0A0%0D%0A%0D%0Ax2+%0D%0A𝑥%0D%0A2%0D%0A+%0D%0Ax+%0D%0A𝑥%0D%0A+%0D%0Ax+42=0

Question

One of the steps needed to solve a quadratic equation by grouping is to split the middle term as a sum of two terms. Given the equation  x2−13x+42=0%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A13%0D%0A𝑥%0D%0A+%0D%0A42%0D%0A=%0D%0A0%0D%0A , which two numbers would be needed to complete this step? Enter the lesser value first.(1 point)%0D%0Ax2−13x+42=0%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A13%0D%0A𝑥%0D%0A+%0D%0A42%0D%0A=%0D%0A0%0D%0A%0D%0Ax2+%0D%0A𝑥%0D%0A2%0D%0A+%0D%0Ax+%0D%0A𝑥%0D%0A+%0D%0Ax+42=0

GPT-4o mini · Answer

To solve the quadratic equation $ x^2 - 13x + 42 = 0 $ by grouping, we need to split the middle term, which is $-13x$, into two terms that add up to $-13$ and multiply to $42$. We are looking for two numbers $a$ and $b$ such that: 1. $ a + b = -13 $ 2. $ a \cdot b = 42 $ Let's find pairs of factors of $42$: - $1$ and $42$ - $2$ and $21$ - $3$ and $14$ - $6$ and $7$ Now let's consider the sums of these factor pairs: - $1 + 42 = 43$ - $2 + 21 = 23$ - $3 + 14 = 17$ - $6 + 7 = 13$ Since we need them to sum to $-13$, we can take the negatives of each: - $-1$ and $-42$ (Sum = -43) - $-2$ and $-21$ (Sum = -23) - $-3$ and $-14$ (Sum = -17) - $-6$ and $-7$ (Sum = -13) The pair $-6$ and $-7$ successfully sums to $-13$ and multiplies to $42$. Therefore, the two numbers that complete the grouping step are $-6$ and $-7$. **Answer**: -7, -6