Difference between revisions of "1950 AHSME Problems/Problem 38"

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By <math> (acdb)=ab-cd </math>, we have <math>2x^2-x=3</math>. Subtracting <math>3</math> from both sides, giving <math>2x^2-x-3=0</math>. This factors to <math>(2x-3)(x+1)=0</math>. Thus, <math>x=\dfrac{3}{2},-1</math>, so the equation is <math>\boxed{\textbf{(B)}\ \text{satisified for only 2 values of }x}</math>.
 
By <math> (acdb)=ab-cd </math>, we have <math>2x^2-x=3</math>. Subtracting <math>3</math> from both sides, giving <math>2x^2-x-3=0</math>. This factors to <math>(2x-3)(x+1)=0</math>. Thus, <math>x=\dfrac{3}{2},-1</math>, so the equation is <math>\boxed{\textbf{(B)}\ \text{satisified for only 2 values of }x}</math>.
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Note: Alternatively, one may note that the equation is quadratic with a nonzero discriminant, so it will be satisfied for exactly two values of <math>x</math>.
  
 
==See Also==
 
==See Also==
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[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 00:35, 20 January 2024

Problem

If the expression $\begin{pmatrix}a & c\\ d & b\end{pmatrix}$ has the value $ab-cd$ for all values of $a, b, c$ and $d$, then the equation $\begin{pmatrix}2x & 1\\ x & x\end{pmatrix}= 3$:

$\textbf{(A)}\ \text{Is satisfied for only 1 value of }x\qquad\\ \textbf{(B)}\ \text{Is satisified for only 2 values of }x\qquad\\ \textbf{(C)}\ \text{Is satisified for no values of }x\qquad\\ \textbf{(D)}\ \text{Is satisfied for an infinite number of values of }x\qquad\\ \textbf{(E)}\ \text{None of these.}$

Solution

By $\begin{pmatrix}a & c\\ d & b\end{pmatrix}=ab-cd$, we have $2x^2-x=3$. Subtracting $3$ from both sides, giving $2x^2-x-3=0$. This factors to $(2x-3)(x+1)=0$. Thus, $x=\dfrac{3}{2},-1$, so the equation is $\boxed{\textbf{(B)}\ \text{satisified for only 2 values of }x}$.

Note: Alternatively, one may note that the equation is quadratic with a nonzero discriminant, so it will be satisfied for exactly two values of $x$.

See Also

1950 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 37
Followed by
Problem 39
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