Question

Решите уравнение используя способ введения новой переменной (у^2-8)^2+2(y^2-8)-8=0 (Помогите)

Answer 1

${y}^{4} - 16 {y}^{2} + 64 + 2 {y}^{2} - 16 - 8 = 0 \\ {y}^{4} - 14 {y}^{2} + 40 = 0$
замена y²=t
$t \geqslant 0$
$t1.2 = \frac{14 + - 6}{2}$
$t1 = 10 \\ t2 = 4$
${y}^{2} = 10 \\ y =+- \sqrt{10}$
${y}^{2} = 4 \\ y = +-2$

Answer 2

$y^2-8=t;\ t^2+2t-8=0;\ (t+4)(t-2)=0;\ \left [ {{t=-4} \atop {t=2}} \right.$

$\left [ {{y^2-8+4=0} \atop {y^2-8-2=0}} \right. \Leftrightarrow \left [ {{y=\pm 2} \atop {y=\pm\sqrt{10}}} \right.$

Answer 3

Solving the Equation Using a New Variable

To solve the equation *(y^2-8)^2 + 2(y^2-8) - 8 = 0* using the method of introducing a new variable, we can let *u = y^2 - 8*. This substitution simplifies the equation to *u^2 + 2u - 8 = 0*.

Quadratic Equation Solution

To solve the quadratic equation *u^2 + 2u - 8 = 0*, we can use the quadratic formula: u = (-b ± √(b^2 - 4ac)) / (2a), where *a = 1*, *b = 2*, and *c = -8*.

Applying the Quadratic Formula

Substituting the values of *a*, *b*, and *c* into the quadratic formula: u = (-2 ± √(2^2 - 4*1*(-8))) / (2*1)

Calculating the Solutions

Solving for *u*: u = (-2 ± √(4 + 32)) / 2 u = (-2 ± √36) / 2 u = (-2 ± 6) / 2

This gives us two possible values for *u*: u₁ = (-2 + 6) / 2 = 2 u₂ = (-2 - 6) / 2 = -4

Substituting Back for y

Now that we have the values of *u*, we can substitute back to find the values of *y*: For *u = 2*: y^2 - 8 = 2 y^2 = 10 y = ±√10

For *u = -4*: y^2 - 8 = -4 y^2 = 4 y = ±2

Therefore, the solutions for the original equation are: y = ±√10 and y = ±2.