Difference between revisions of "1955 AHSME Problems/Problem 12"
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<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> | <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 | After that, adding <math>2</math> to both sides will give us | ||
<cmath>2\cdot\sqrt{5x^2-6x+1}-2+2=-6x+4+2 \Longrightarrow 2\cdot\sqrt{5x^2-6x+1}=-6x+6</cmath> | <cmath>2\cdot\sqrt{5x^2-6x+1}-2+2=-6x+4+2 \Longrightarrow 2\cdot\sqrt{5x^2-6x+1}=-6x+6</cmath> | ||
+ | Next, we divide both sides by 2 which gives us | ||
− | + | <cmath>\frac{2\cdot\sqrt{5x^2-6x+1}-2+2}{2}=\frac{-6x+4+2}{2} \Longrightarrow \sqrt{5x^2-6x+1}=-3x+3</cmath> | |
+ | Finally, solving the equation, we get | ||
+ | |||
+ | <cmath>5x^2-6x+1=(-3x+3)^2 \Longrightarrow 5x^2-6x+1=9x^2-18x+9</cmath> | ||
+ | <cmath>\Longrightarrow 5x^2-6x+1-(9x^2-18x+9)=9x^2-18x+9-(9x^2-18x+9) </cmath> | ||
+ | <cmath>\Longrightarrow -4x^2+12x-8=0 \Longrightarrow -4(x-1)(x-2)=0</cmath> | ||
+ | <cmath>\Longrightarrow x-1=0 \text{or}\ x-2=0 \Longrightarrow x=1 \text{or}\ x=2</cmath> | ||
+ | Plugging 1 and 2 into the original equation, <math>\sqrt{5x-1}+\sqrt{x-1}=2</math>, we see that when <math>x=1</math> | ||
+ | |||
+ | <cmath>\sqrt{5x-1}+\sqrt{x-1}=2 \Longrightarrow \sqrt{5\cdot1-1}+\sqrt{1-1}=2 \Longrightarrow \sqrt4+\sqrt0=2 \Longrightarrow 2+0=2 \Longrightarrow 2=2</cmath> | ||
+ | the equation is true. On the other hand, we note that when <math>x=2</math> | ||
− | <math></math> | + | <cmath>\sqrt{5x-1}+\sqrt{x-1}=2 \Longrightarrow \sqrt{5\cdot2-1}+\sqrt{2-1}=2 \Longrightarrow \sqrt9+\sqrt1=2 \Longrightarrow 3+1=2 \Longrightarrow 4=0</cmath> |
+ | the equation is false. | ||
+ | Therefore the answer is <math>\fbox{{\bf(D)} x=1}</math> |
Revision as of 01:36, 10 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
Next, we divide both sides by 2 which gives us
Finally, solving the equation, we get
Plugging 1 and 2 into the original equation,
, we see that when
the equation is true. On the other hand, we note that when
the equation is false.
Therefore the answer is