Question

(8x-3)²- 10|8x-3| -24=0

(x² +2x+2) (x²+2x-4)=-5

Answer 1

$(8x-3)^{2} -10|8x-3|-24=0\\\\|8x-3|^{2}-10|8x-3|-24=0\\\\\left[\begin{array}{ccc}|8x-3|=12\\|8x-3|=-2-\oslash\end{array}\right \\\\|8x-3|=12\\\\\left[\begin{array}{ccc}8x-3=12\\8x-3=-12\end{array}\right\\\\\\\left[\begin{array}{ccc}x=1,875\\x=-1,125\end{array}\right$

(x² + 2x + 2)(x² + 2x - 4) = - 5

Сделаем замену :

x² + 2x - 4 = m   ⇒  x² + 2x + 2 = m + 6

(m + 6) * m + 5 = 0

m² + 6m + 5 = 0

По теореме Виета :

m₁ = - 1        m₂ = - 5

1)  x² + 2x - 4 = - 1

x² + 2x - 3 = 0

x₁ = - 3       x₂ = 1

2) x² + 2x - 4 = - 5

x² + 2x + 1 = 0

(x + 1)² = 0

x + 1 = 0

x₃ = -1

Ответ : - 3 ; -1 ; 1

Answer 2

$(8x-3)^2-10|8x-3|-24=0\Leftrightarrow |8x-3|^2-10|8x-3|-24=0\\D_{|8x-3|}=25+24=49\\|8x-3|=5\pm 7\Leftrightarrow |8x-3|=12\Leftrightarrow (8x-3-12)(8x-3+12)=0\Leftrightarrow \\\Leftrightarrow (8x-15)(8x+9)=0\Rightarrow x=\begin{Bmatrix}-\frac{9}{8};\frac{15}{8}\end{Bmatrix}$

$(x^2+2x+2)(x^2+2x-4)=-5\Leftrightarrow (x^2+2x)^2-2(x^2+2x)-3=0\\ax^2+bx+c=0\\a-b+c=0\Rightarrow x=\begin{Bmatrix}-1;-\frac{c}{a}\end{Bmatrix}\\1+2-3=0\Rightarrow x^2+2x=\begin{Bmatrix}-1;3\end{Bmatrix}\\x^2+2x+1=0\Leftrightarrow (x+1)^2=0\Rightarrow x=-1\\x^2+2x-3=0\Rightarrow x=\begin{Bmatrix}1;-3\end{Bmatrix}\\x=\begin{Bmatrix}-3;-1;1\end{Bmatrix}$