The smaller root of the equation &(x-34)(x-34) & +(x-34) (x-12)=0 is:

1964 AMC 12 Problem 8

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The smaller root of the equation

\begin{aligned} &\left(x-\frac34\right)\left(x-\frac34\right)\\ &\quad+\left(x-\frac34\right) \left(x-\frac12\right)=0 \end{aligned}

is:

-\dfrac34

\dfrac12

\dfrac58

\dfrac34

1

Factor out the common expression

x-\frac34

The remaining linear factor is

2x-\frac54

Combining the two terms and factoring gives

\left(x-\frac34\right)\left(2x-\frac54\right)=0.

Hence the roots are

\frac{3}{4}

and

\frac{5}{8},

of which

\frac{5}{8}

is smaller.

Therefore, the correct answer is C.