Algebraic Identity Identification
The expression $a^2 + b^2 + c^2 - 2ab + 2ac - 2bc$ is recognized as the expansion of a squared trinomial. Specifically, it matches the form $(a - b + c)^2$.
Verification: $(a - b + c)^2 = a^2 + (-b)^2 + c^2 + 2(a)(-b) + 2(a)(c) + 2(-b)(c)$
$= a^2 + b^2 + c^2 - 2ab + 2ac - 2bc$.
Therefore, the given expression is equivalent to $(a - b + c)^2$.
Substitution of Values
Substitute the provided values for $a$, $b$, and $c$ into the equivalent expression $(a - b + c)^2$:
$a = x + y$
$b = x - y$
$c = 2x - 1$
Substituting these into $(a - b + c)^2$ yields: $((x + y) - (x - y) + (2x - 1))^2$
Expression Simplification
Simplify the terms inside the parentheses:
$(x + y) - (x - y) + (2x - 1)$
Distribute the negative sign: $x + y - x + y + 2x - 1$
Combine like terms: $(x - x + 2x) + (y + y) - 1$
$= 2x + 2y - 1$
Final Result Calculation
The simplified expression inside the parentheses is $2x + 2y - 1$. Squaring this result provides the final value:
$(2x + 2y - 1)^2$
Similar Questions
- If $a = 9$, $b = -5$ and $c = -4$, then find the value of $a^{3} + b^{3} + c^{3}$.
- If $x^2 = y+z, y^2 = z+x$ and $z^2 = x+y$, then find the value of $\frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1}$.
- If $x^4 + \frac{1}{x^4} = 322$ and $x > 1$, then what is the value of $x^3 - \frac{1}{x^3}$?
- If $x - \frac{1}{x} = 5$, find the value of $x^4 + \frac{1}{x^4}$.
- If $x + \frac{1}{x} = 2$, then the value of $x^{2} + \frac{1}{x^{2}} = ?$
- If $x + y + z = 0$, then what will be the value of $\frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy}$ ?
- If $x^3 + \frac{1}{x^3} = 18$, then what will be the value of $x + \frac{1}{x}$?
- If $x + \frac{1}{x} = 5$
Then the value of $\frac{3x}{2x^2 + 2 - 5x}$ will be: - If $3a + 4b = 2$ and $ab = \frac{1}{36}$, then $27a^3 + 64b^3$ is:
- If $a^2 + b^2 + c^2 + 3 = 2(a + b + c)$, then the value of $(a + b + c)$ is:
Important Questions from Algebra
If 2x – y = 2 and xy = \(\frac{3}{2}\) , then what is the value of x 3– \(\frac{{{y^3}}}{8}\) ?
If (10a 3+ 4b 3) : (11a 3- 15b 3) = 7 : 5, then (3a + 5b) : (9a - 2b) =?
- If 4sin 2 θ = 3(1+ cos θ), 0° < θ < 90°, then what is the value of (2tan θ + 4sin θ - sec θ)?
The value of:
\(\frac{{\sin 23^\circ \cos 67^\circ + \sec52^\circ \sin38^\circ + \cos 23^\circ \sin 67^\circ + \rm cosec52^\circ \cos 38^\circ }}{{\rm cose{c^2}20^\circ - {{\tan }^2}70^\circ }}\)
If (x + y) 3+ 27(x - y) 3= (Ax - 2y)(Bx 2+ Cxy + 13y 2), then the value of A - B - C is:
