Solving the Algebraic Equation for x
The problem asks for the value of $x$ in the equation: $ \left(x-\frac{1}{2}\right)^2 - \left(x-\frac{3}{2}\right)^2 = x+2 $ We can simplify the left side of the equation using the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$.
Applying the Difference of Squares Formula
Let $a = \left(x-\frac{1}{2}\right)$ and $b = \left(x-\frac{3}{2}\right)$.
- Calculate $a-b$: $ a-b = \left(x-\frac{1}{2}\right) - \left(x-\frac{3}{2}\right) = x - \frac{1}{2} - x + \frac{3}{2} = \frac{2}{2} = 1 $
- Calculate $a+b$: $ a+b = \left(x-\frac{1}{2}\right) + \left(x-\frac{3}{2}\right) = 2x - \left(\frac{1}{2} + \frac{3}{2}\right) = 2x - \frac{4}{2} = 2x - 2 $
Now substitute these back into the difference of squares formula:
$ a^2 - b^2 = (a-b)(a+b) = (1)(2x-2) = 2x-2 $Solving for x
Equate the simplified left side to the right side of the original equation:
$ 2x-2 = x+2 $Rearrange the terms to solve for $x$:
- Subtract $x$ from both sides: $ 2x - x - 2 = x - x + 2 $ $ x - 2 = 2 $
- Add 2 to both sides: $ x - 2 + 2 = 2 + 2 $ $ x = 4 $
Thus, the value of $x$ is 4.
Important Questions from Algebra
- If $Pe^x = Qe^{-x}$ for all real values of $x$, which one of the following statements is true?
The relationship between two variables $x$ and $y$ is given by $x + py + q = 0$ and is shown in the figure. Find the values of $p$ and $q$.
Note: The figure shown is representative.
The real variables $x, y, z$ and the real constants $p, q, r $ satisfy
$\frac{x}{pq - r^2} = \frac{y}{qr - p^2} = \frac{z}{rp - q^2}$
Given the denominators are non-zero, the value of $px + qy + rz$ isThe complex function
$e^{-\left(\frac{2}{z-1}\right)}$
has __________________Consider two matrices: $P = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $Q = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$.
Which of the following statement is/are true?
