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Difference between revisions of "1955 AHSME Problems/Problem 19"

(Created page with "== Problem 19== Two numbers whose sum is <math>6</math> and the absolute value of whose difference is <math>8</math> are roots of the equation: <math> \textbf{(A)}\ x^2-6x+...")
 
 
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Expanding the above equation gets us <math>\textbf{(B)} x^2 - 6x - 7 = 0</math>
 
Expanding the above equation gets us <math>\textbf{(B)} x^2 - 6x - 7 = 0</math>
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==Solution 2==
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Let the roots of the equation be <math>x</math> and <math>y</math>. Therefore, we can set up a system of equation:
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<cmath>x+y=6</cmath><cmath>|x-y|=8</cmath>Therefore, we get <math>x=7</math> and <math>y=-1</math>. So, <math>(x-7)(x+1)=\boxed{x^2-6x-7}</math>
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- kante314 -
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== See Also ==
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{{AHSME box|year=1955|num-b=18|num-a=20}}
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{{MAA Notice}}

Latest revision as of 10:11, 12 July 2021

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 -

See Also

1955 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions


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