1963 AHSME Problems/Problem 4

Problem 4

For what value(s) of $k$ does the pair of equations $y=x^2$ and $y=3x+k$ have two identical solutions?

$\textbf{(A)}\ \frac{4}{9}\qquad \textbf{(B)}\ -\frac{4}{9}\qquad \textbf{(C)}\ \frac{9}{4}\qquad \textbf{(D)}\ -\frac{9}{4}\qquad \textbf{(E)}\ \pm\frac{9}{4}$

Solution

If the system of equations has two identical solutions, then only one $(x,y)$ pair will satisfy both equations.

Substitute $y$ in one equation into another equation. $x^2 = 3x + k$ $x^2 - 3x = k$ Complete the square to get $x^2 - 3x + \frac{9}{4} = k + \frac{9}{4}$ $(x - \frac{3}{2})^2 = k + \frac{9}{4}$ In order for the equation to have one solution, the right side must be $0$ , so $k = -\frac{9}{4}$ , which is answer choice $\boxed{\textbf{(D)}}$ .

1963 AHSC (Problems • Answer Key • Resources)
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1963 AHSME Problems/Problem 4

Problem 4

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