تمارين مع الإصلاح
C1: \( (2x + 1)^2 - (3x - 2)(3x + 2) \)
الإصلاح:
\[
\begin{align*}
(2x + 1)^2 - (3x - 2)(3x + 2) &= 4x^2 + 4x + 1 - [9x^2 - 4] \\
&= 4x^2 + 4x + 1 - 9x^2 + 4 \\
&= -5x^2 + 4x + 5
\end{align*}
\]
C2: \( (2x + 3)^2 - (2x + 3)(x - 7) \)
الإصلاح:
\[
\begin{align*}
(2x + 3)^2 - (2x + 3)(x - 7) &= 4x^2 + 12x + 9 - [2x^2 - 14x + 3x - 21] \\
&= 4x^2 + 12x + 9 - 2x^2 + 14x - 3x + 21 \\
&= 2x^2 + 23x + 30
\end{align*}
\]
C3: \( 3(7x - 4)^2 - (x + 2)^2 \)
الإصلاح:
\[
\begin{align*}
3(7x - 4)^2 - (x + 2)^2 &= 3(49x^2 - 56x + 16) - [x^2 + 4x + 4] \\
&= 147x^2 - 168x + 48 - x^2 - 4x - 4 \\
&= 146x^2 - 172x + 44
\end{align*}
\]
C4: \( (7x - 2)(3x + 1) - 2(2x + 1)^2 \)
الإصلاح:
\[
\begin{align*}
(7x - 2)(3x + 1) - 2(2x + 1)^2 &= 21x^2 + 7x - 6x - 2 - 2[4x^2 + 4x + 1] \\
&= 21x^2 + x - 2 - 8x^2 - 8x - 2 \\
&= 13x^2 - 7x - 4
\end{align*}
\]
C5: \( 5(3x - 9)(3x + 9) - (3x + 4)^2 \)
الإصلاح:
\[
\begin{align*}
5(3x - 9)(3x + 9) - (3x + 4)^2 &= 5[9x^2 - 81] - [9x^2 + 24x + 16] \\
&= 45x^2 - 405 - 9x^2 - 24x - 16 \\
&= 36x^2 - 24x - 421
\end{align*}
\]
C6: \( (2x - 1)^2 - (2x + 3)(2x - 5) \)
الإصلاح:
\[
\begin{align*}
(2x - 1)^2 - (2x + 3)(2x - 5) &= 4x^2 - 4x + 1 - [4x^2 - 10x + 6x - 15] \\
&= 4x^2 - 4x + 1 - 4x^2 + 10x - 6x + 15 \\
&= 16
\end{align*}
\]
C7: \( (3x + 1)^2 - 2(3x - 1)^2 \)
الإصلاح:
\[
\begin{align*}
(3x + 1)^2 - 2(3x - 1)^2 &= 9x^2 + 6x + 1 - 2[9x^2 - 6x + 1] \\
&= 9x^2 + 6x + 1 - 18x^2 + 12x - 2 \\
&= -9x^2 + 18x - 1
\end{align*}
\]
C8: \( 2(2 - 14x)^2 - (7x - 1)^2 \)
الإصلاح:
\[
\begin{align*}
2(2 - 14x)^2 - (7x - 1)^2 &= 2(4 - 56x + 196x^2) - [49x^2 - 14x + 1] \\
&= 8 - 112x + 392x^2 - 49x^2 + 14x - 1 \\
&= 343x^2 - 98x + 7
\end{align*}
\]
C9: \( 2x(x - 5) - (x + 1)^2 \)
الإصلاح:
\[
\begin{align*}
2x(x - 5) - (x + 1)^2 &= 2x^2 - 10x - [x^2 + 2x + 1] \\
&= 2x^2 - 10x - x^2 - 2x - 1 \\
&= x^2 - 12x - 1
\end{align*}
\]
C10: \( (x + 3)^2 - (2x + 1)(2x - 1) \)
الإصلاح:
\[
\begin{align*}
(x + 3)^2 - (2x + 1)(2x - 1) &= x^2 + 6x + 9 - [4x^2 - 1] \\
&= x^2 + 6x + 9 - 4x^2 + 1 \\
&= -3x^2 + 6x + 10
\end{align*}
\]
C11: \( 5x^2 - (x + 3)(x - 3) \)
الإصلاح:
\[
\begin{align*}
5x^2 - (x + 3)(x - 3) &= 5x^2 - [x^2 - 9] \\
&= 5x^2 - x^2 + 9 \\
&= 4x^2 + 9
\end{align*}
\]
C12: \( (3x + 2)^2 - (x - 1)^2 \)
الإصلاح:
\[
\begin{align*}
(3x + 2)^2 - (x - 1)^2 &= 9x^2 + 12x + 4 - [x^2 - 2x + 1] \\
&= 9x^2 + 12x + 4 - x^2 + 2x - 1 \\
&= 8x^2 + 14x + 3
\end{align*}
\]
C13: \( 5(x - 1)(x + 2) - (x + 3)^2 \)
الإصلاح:
\[
\begin{align*}
5(x - 1)(x + 2) - (x + 3)^2 &= 5(x^2 + 2x - x - 2) - [x^2 + 6x + 9] \\
&= 5x^2 + 10x - 5x - 10 - x^2 - 6x - 9 \\
&= 4x^2 - x - 19
\end{align*}
\]
C14: \( (4x - 3)^2 + 7(x - 2)(x - 5) \)
الإصلاح:
\[
\begin{align*}
(4x - 3)^2 + 7(x - 2)(x - 5) &= 16x^2 - 24x + 9 + 7[x^2 - 5x - 2x + 10] \\
&= 16x^2 - 24x + 9 + 7x^2 - 35x - 14x + 70 \\
&= 23x^2 - 73x + 79
\end{align*}
\]
C15: \( -3(x - 6)^2 - 7(3x - 4)(3x + 4) \)
الإصلاح:
\[
\begin{align*}
-3(x - 6)^2 - 7(3x - 4)(3x + 4) &= -3[x^2 - 12x + 36] - 7[9x^2 - 16] \\
&= -3x^2 + 36x - 108 - 63x^2 + 112 \\
&= -66x^2 + 36x + 4
\end{align*}
\]