# 1955 AHSME Problems/Problem 19

## Problem 19

Two numbers whose sum is $6$ and the absolute value of whose difference is $8$ are roots of the equation: $\textbf{(A)}\ x^2-6x+7=0\qquad\textbf{(B)}\ x^2-6x-7=0\qquad\textbf{(C)}\ x^2+6x-8=0\\ \textbf{(D)}\ x^2-6x+8=0\qquad\textbf{(E)}\ x^2+6x-7=0$

## Solution

The first two hints can be expressed as the following system of equations: $$\begin{cases} (1) & a + b = 6 \\ (2) & a - b = 8 \end{cases}$$ From this, we can clearly see that $a = 7$, and that $b = -1$.

Since quadratic equations can generally be expressed in the form of $(x - a)(x - b) = 0$, where a and b are roots, the correct quadratic, once factored, would look like $(x - 7)(x + 1) = 0$

Expanding the above equation gets us $\textbf{(B)} x^2 - 6x - 7 = 0$

## Solution 2

Let the roots of the equation be $x$ and $y$. Therefore, we can set up a system of equation: $$x+y=6$$ $$|x-y|=8$$Therefore, we get $x=7$ and $y=-1$. So, $(x-7)(x+1)=\boxed{x^2-6x-7}$

- kante314 -

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