Bonjour,
a.
[tex](x+1)^2 = 4\\(x+1)^2-4=0\\(x+1)^2 - 2^2 = 0\\ \rightarrow a^2 - b^2 = (a-b)(a+b)\\
(x+1-2)(x+1+2) = 0\\
x-1 = 0 \quad \text{ou}\quad x+3 = 0\\
x = 1 \quad \text{ou} \quad x = -3\\\\
\boxed{S = \left\{-3; 1\right\}}[/tex]
b.
[tex](2x+1)^2 = x^2\\ (2x+1)^2 - x^2 = 0\\\rightarrow a^2 - b^2 = (a-b)(a+b)\\ (2x+1-x)(2x+1+x)=0\\ (x+1)(3x+1) = 0\\ x = -1 \quad \text{ou} \quad 3x = -1\\\\ x = -1 \quad \text{ou} \quad x = -\dfrac{1}{3}\\\\ \boxed{S = \left\{-1; -\dfrac{1}{3}\right\}}[/tex]
c.
[tex]x (2x+1) = 2x\\
x(2x+1)-2x = 0\\
(2x+1-2)x = 0\\
(2x-1)x = 0\\
2x-1 = 0 \quad \text{ou} \quad x = 0\\
2x = 1 \quad \text{ou} \quad x = 0\\\\
x = \dfrac{1}{2} \quad \text{ou} \quad x = 0\\\\
\boxed{S = \left\{0; \dfrac{1}{2}\right\}}[/tex]
d.
[tex](-3x+1)^2 = 25\\
(-3x+1)^2 - 25=0\\
(-3x+1)^2 -5^2=0\\
\rightarrow a^2 - b^2 = (a-b)(a+b)\\
(-3x+1-5)(-3x+1+5)=0\\
-3x-4=0 \quad \text{ou} \quad -3x+6=0\\
3x = -4 \quad \text{ou} \quad 3x = 6\\\\
x = -\dfrac{4}{3} \quad \text{ou} \quad x = \dfrac{6}{3} = 2\\\\
\boxed{S = \left\{-\dfrac{4}{3}; 2\right\}}[/tex]
e.
[tex](4x-1)(5x-2)-16x^2+1=0\\
(4x-1)(5x-2)+1-16x^2=0\\\\
(4x-1)(5x-2)-1^2-(4x)^2=0\\
\rightarrow a^2 - b^2 = (a-b)(a+b)\\
(4x-1)(5x-2)+(1-4x)(1+4x)=0\\
(4x-1)(5x-2)-(4x-1)(1+4x)\\
(4x-1)(5x-2-(1+4x))\\
(4x-1)(x-3)\\
4x-1 = 0 \quad \text{ou} \quad x-3 = 0\\
4x=1 \quad \text{ou} \quad x = 3\\\\
x = \dfrac{1}{4} \quad \text{ou} \quad x = 3\\\\
\boxed{S = \left\{\dfrac{1}{4}; 3\right\}}
[/tex]
f.
[tex]\dfrac{9}{4}x^2+3x+1=0\\\\
\left(\dfrac{9}{4}x^2+3x+1\right)\times 4=0\times 4\\\\
9x^2+12x+4 = 0\\
\rightarrow a^2+2ab+b^2 = (a+b)^2\\
(3x)^2+2\times3x\times2 + 2^2\\
(3x+2)^2=0\\
3x+2 = 0\\
3x = -2\\\\
x = -\dfrac{2}{3}\\\\
\boxed{S = \left\{-\dfrac{2}{3}\right\}}[/tex]
g.
[tex]6x+4=(2x-5)(9x+6)\\
2(3x+2) = (2x-5)\times3(3x+2)\\\\
2(3x+2)-(2x-5)\times 3(3x+2) = 0\\
(3x+2)(2-(2x-5)\times3)=0\\
(3x+2)(2-6x+15)=0\\
(3x+2)(17-6x) =0\\
3x+2 = 0 \quad \text{ou} \quad 17-6x=0\\\\
3x = -2 \quad \text{ou} \quad 6x = 17\\\\
x = -\dfrac{2}{3} \quad \text{ou} \quad x = \dfrac{17}{6}\\\\
\boxed{S = \left\{-\dfrac{2}{3}; \dfrac{17}{6}\right\}}[/tex]
h.
[tex]x^2+25=10x\\
x^2+25-10x=0\\
x^2-10x+25=0\\
\rightarrow a^2-2ab+b^2 = (a-b)^2\\
x^2 - 2\times x \times 5 \times 5^2=0\\
(x-5)^2 = 0
x-5 = 0\\
x = 5\\\\
\boxed{S = \left\{5\right\}}[/tex]
