Difference between revisions of "1955 AHSME Problems/Problem 12"
Awesomechoco (talk | contribs) (Created page with "==Problem== The solution of <math>\sqrt{5x-1}+\sqrt{x-1}=2</math> is: <math>\textbf{(A)}\ x=2,x=1\qquad\textbf{(B)}\ x=\frac{2}{3}\qquad\textbf{(C)}\ x=2\qquad\textbf{(D)}\ x...") |
Awesomechoco (talk | contribs) (→Solution) |
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<math>\textbf{(A)}\ x=2,x=1\qquad\textbf{(B)}\ x=\frac{2}{3}\qquad\textbf{(C)}\ x=2\qquad\textbf{(D)}\ x=1\qquad\textbf{(E)}\ x=0</math> | <math>\textbf{(A)}\ x=2,x=1\qquad\textbf{(B)}\ x=\frac{2}{3}\qquad\textbf{(C)}\ x=2\qquad\textbf{(D)}\ x=1\qquad\textbf{(E)}\ x=0</math> | ||
− | ==Solution== | + | ==Solution 1== |
− | First, square both sides. | + | First, square both sides. This gives us |
− | < | + | <cmath>\sqrt{5x-1}^2+2\cdot\sqrt{5x-1}\cdot\sqrt{x-1}+\sqrt{x-1}^2=4 \Longrightarrow 5x-1+2\cdot\sqrt{(5x-1)\cdot(x-1)}+x-1=4 \Longrightarrow 2\cdot\sqrt{5x^2-6x+1}+6x-2=4</cmath> |
− | Then, | + | Then, adding <math>-6x</math> to both sides gives us |
− | < | + | <cmath>2\cdot\sqrt{5x^2-6x+1}+6x-2-6x=4-6x \Longrightarrow 2\cdot\sqrt{5x^2-6x+1}-2 =-6x+4</cmath> |
+ | |||
+ | After that, adding <math>2</math> to both sides will give us | ||
+ | |||
+ | <cmath>2\cdot\sqrt{5x^2-6x+1}-2+2=-6x+4 \Longrightarrow 2\cdot\sqrt{5x^2-6x+1}=-6x-2</cmath> |
Revision as of 23:21, 9 July 2018
Problem
The solution of is:
Solution 1
First, square both sides. This gives us
Then, adding
to both sides gives us
After that, adding to both sides will give us