An algebra problem by Armain Labeeb

Algebra Level 2

Cecilia attempts to prove that 0 = 1 0\,=\,1 . This was her proof:

− 20 = − 20 16 − 36 = 25 − 45 4 2 − 4 ⋅ 9 = 5 2 − 5 ⋅ 9 4 2 − 4 ⋅ 9 + 81 4 = 5 2 − 5 ⋅ 9 + 81 4 4 2 − 2 ⋅ 4 ⋅ 9 2 + ( 9 2 ) 2 = 5 2 − 5 ⋅ 9 + ( 9 2 ) 2 ( 4 − 9 2 ) 2 = ( 5 − 9 2 ) 2 ( 4 − 9 2 ) = ( 5 − 9 2 ) ( 4 − 9 2 ) + 1 2 = ( 5 − 9 2 ) + 1 2 0 = 1 \begin{aligned} -20 & \, \, =\, \, -20 \\ 16\, -\, 36 & \, \, =\, \, 25\, -\, 45 \\ 4^{ 2 }\, -\, 4\, \cdot \, 9 & \, \, =\, \, 5^{ 2 }\, -\, 5\, \cdot \, 9 \\ 4^{ 2 }\, -\, 4\, \cdot \, 9\, +\, \frac { 81 }{ 4 } & \, \, =\, \, 5^{ 2 }\, -\, 5\, \cdot \, 9\, +\, \frac { 81 }{ 4 } \\ 4^{ 2 }\, -\, 2\, \cdot \, 4\, \cdot \, \frac { 9 }{ 2 } \, +\, \left( \frac { 9 }{ 2 } \right) ^{ 2 }\, \, & \, \, =\, \, 5^{ 2 }\, -\, 5\, \cdot \, 9\, +\, \, \left( \frac { 9 }{ 2 } \right) ^{ 2 } \\ \left( 4\, -\, \frac { 9 }{ 2 } \right) ^{ 2 }\, \, & =\, \, \left( 5\, -\, \frac { 9 }{ 2 } \right) ^{ 2 } \\ \left( 4\, -\, \frac { 9 }{ 2 } \right) \, \, & =\, \, \left( 5\, -\, \frac { 9 }{ 2 } \right) \\ \left( 4\, -\, \frac { 9 }{ 2 } \right) \, +\, \frac { 1 }{ 2 } \, \, & =\, \, \left( 5\, -\, \frac { 9 }{ 2 } \right) \, +\, \frac { 1 }{ 2 } \\ 0 & \, \, =\, \, 1 \end{aligned}

Which is the first step where she made a mistake? Assume that the first line of the equation ( − 20 = − 20 -20=-20 ) is the first step.

Signs: a ⋅ b ⋅ c = ( a × b × c ) a\,\cdot\,b\,\cdot\,c\,=\,(a\,\times\,b\,\times\,c) .

8 9 4 5 2 3 6 7

Armain Labeeb by Armain Labeeb , 20, Bangladesh

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1 solution

The square root function is defined as x ² = ∣ x ∣ 4 − 9 2 < 0 , So we actually have ( 4 − 9 2 ) ² = ∣ 4 − 9 2 ∣ = ∣ − 1 2 ∣ = + 1 2 = 9 2 − 4 Therefore, the mistake occurs in the 7th step where Cecilia wrongly evaluates ( 4 − 9 2 ) ² as 4 − 9 2 \text{The square root function is defined as}\;\sqrt{x²}=|x|\\ \color{#20A900}{4-\frac{9}{2}<0},\text{So we actually have}\;\sqrt{\left(4-\frac{9}{2}\right)²}=\left|4-\frac{9}{2}\right|=\left|-\frac{1}{2}\right|=+\frac{1}{2}=\color{#3D99F6}{\frac{9}{2}-4}\\ \text{Therefore, the mistake occurs in the 7th step where Cecilia wrongly evaluates}\;\sqrt{\left(4-\frac{9}{2}\right)²}\;\text{as}\; 4-\frac{9}{2}

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