Check whether the following are : (i) (x – 2)2 + 1 = 2x – 3 (ii) x(x + 1) + 8 = (x + 2) (x – 2) (iii) x (2x + 3) = x2 + 1 (iv) (x + 2)3 = x3 – 4
Step-by-Step Solution
Key Concept: To verify whether an equation is an identity, expand both sides, bring all terms to one side and simplify. If the resulting expression reduces to 0 (or a true statement) for <b>all</b> values of \(x\), the equation is an identity. If it holds only for particular values of \(x\) or never, it is not an identity.
1. Equation (i)\
\[ (x-2)^2 + 1 = 2x-3 \]
Expand the left side: \( (x-2)^2 = x^2-4x+4 \).\
Hence \( LHS = x^2-4x+4+1 = x^2-4x+5 \).
Bring all terms to one side:\
\[ x^2-4x+5-(2x-3)=0 \Rightarrow x^2-6x+8=0 \]
This quadratic factors as \((x-2)(x-4)=0\), giving \(x=2\) or \(x=4\). Since it is true only for these two values, the given relation is not an identity.
2. Equation (ii)\
\[ x(x+1)+8 = (x+2)(x-2) \]
Expand both sides: \( LHS = x^2 + x + 8 \), \( RHS = x^2 -4 \).
Subtract RHS from LHS:\
\[ x^2 + x + 8 - (x^2 -4)=0 \Rightarrow x + 12 = 0 \]
Hence \( x = -12 \). The equality holds only for \(x=-12\); therefore it is not an identity.
3. Equation (iii)\
\[ x(2x+3) = x^2 + 1 \]
Expand the left side: \( LHS = 2x^2 + 3x \).
Bring all terms to one side:\
\[ 2x^2 + 3x - (x^2 + 1)=0 \Rightarrow x^2 + 3x - 1 = 0 \]
Solving the quadratic gives \( x = \frac{-3 \pm \sqrt{13}}{2} \). The relation is true only for these two values, so it is not an identity.
4. Equation (iv)\
\[ (x+2)^3 = x^3 - 4 \]
Expand the left side using \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\):\
\[ (x+2)^3 = x^3 + 6x^2 + 12x + 8 \]
Bring all terms to one side:\
\[ x^3 + 6x^2 + 12x + 8 - (x^3 - 4)=0 \Rightarrow 6x^2 + 12x + 12 = 0 \]
Divide by 6: \( x^2 + 2x + 2 = 0 \).
The discriminant \( D = 2^2 - 4\cdot1\cdot2 = -4 < 0 \); there is no real solution. Hence the given equation is not an identity (it is never true for real \(x\)).
Summary: None of the four relations are identities; each holds only for specific values of \(x\) (or none in the real domain for (iv)).
Correct Answer: (i) Not an identity (true for \(x=2,4\)).
(ii) Not an identity (true for \(x=-12\)).
(iii) Not an identity (true for \(x = \frac{-3 \pm \sqrt{13}}{2}\)).
(iv) Not an identity (no real \(x\) satisfies it).