I see the exercises in the uploaded image. I'll analyze them one by one and provide solutions along with detailed explanations.

Exercise 5

Expression:

$C = (5x - 3)^2 - (2x - 1)(5x - 3)$

1. Develop, simplify, and order $C$:

   • Expand $(5x - 3)^2 = 25x^2 - 30x + 9$
   • Expand $(2x - 1)(5x - 3) = 10x^2 - 11x + 3$
   • Subtract:
     $C = (25x^2 - 30x + 9) - (10x^2 - 11x + 3)$ $C = 25x^2 - 30x + 9 - 10x^2 + 11x - 3$ $C = 15x^2 - 19x + 6$

2. Calculate $C$ for $x = -2$:

   Substitute $x = -2$ into $C = 15x^2 - 19x + 6$: $C = 15(-2)^2 - 19(-2) + 6$ $C = 15(4) + 38 + 6 = 60 + 38 + 6 = 104$

3. Factorize $C$:

   Factor $15x^2 - 19x + 6$: $C = (3x - 2)(5x - 3)$

4. Solve the equation $(5x - 3)(2 - 3x) = 0$:

   Set each factor to zero:

   • $5x - 3 = 0 \implies x = \frac{3}{5}$
   • $2 - 3x = 0 \implies x = \frac{2}{3}$

   Solutions: $x = \frac{3}{5}, \frac{2}{3}$

Exercise 6

1. Calculate $A = 109 \times 91$:

   Use the difference of squares:
   $109 \times 91 = (100 + 9)(100 - 9) = 100^2 - 9^2 = 10000 - 81 = 9919$

2. Calculate $B = 1252^2 - 752^2$:

   Use the difference of squares:
   $1252^2 - 752^2 = (1252 + 752)(1252 - 752) = 2004 \times 500 = 1002000$

3. Calculate $C = 103^2$:

   Use expansion:
   $103^2 = (100 + 3)^2 = 100^2 + 2(100)(3) + 3^2 = 10000 + 600 + 9 = 10609$

Exercise 7

1. Express the area $A_1$ of rectangle $ABCD$:

   The length of $ABCD$ is $3x - 5$ and the width is $2x + 1$:
   $A_1 = (3x - 5)(2x + 1)$

2. Express the area $A_2$ of polygon $EFGHIJ$:

   Subtract the area of square $FGHI$ (side $3x - 7$) from rectangle $EFGJ$ (dimensions $3x - 7$ and $3x - 5$):
   $A_2 = (3x - 7)(3x - 5) - (3x - 7)^2$ Factorize: $A_2 = (3x - 7)\big((3x - 5) - (3x - 7)\big) = (3x - 7)(2)$ $A_2 = 2(3x - 7)$

3. Solve for $x$ when $A_1 = A_2$:

   Equate $A_1$ and $A_2$:
   $(3x - 5)(2x + 1) = 2(3x - 7)$ Expand and simplify: $6x^2 - 15x + 3x - 5 = 6x - 14$ $6x^2 - 12x + 9 = 0$ Factorize: $3(2x^2 - 4x + 3) = 0$ Solve using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Would you like further details on any step, or clarification? 😊

Related Questions:

1. Can you derive a shortcut for $(a + b)^2 - (a - b)^2$?
2. What are the general steps for factoring quadratics?
3. How do you verify solutions for quadratic equations graphically?
4. Can you solve $(x + 1)^2 = 4$ for real and complex $x$?
5. How do the expressions for area depend on specific values of $x$?

Tip:

When solving quadratic equations, always check whether factoring or the quadratic formula is more efficient!