Question

(y × 5) (y+2)-(y+2)² > 6 решить неравенство​

Answer 1

(5y²+10y)-(y²+4y+4)=6

5y²+10y-y²-4y²-4=6

10y=6+4

10y=10

y=1

Answer 2

Let's solve the inequality step by step:

(y × 5) (y + 2) - (y + 2)² > 6

First, let's simplify the equation:

5y(y + 2) - (y + 2)² > 6

Now, expand and simplify the equation further:

5y^2 + 10y - (y^2 + 4y + 4) > 6

5y^2 + 10y - y^2 - 4y - 4 > 6

4y^2 + 6y - 4 > 6

Next, move all terms to one side of the inequality to get a quadratic equation in standard form:

4y^2 + 6y - 4 - 6 > 0

4y^2 + 6y - 10 > 0

Now, let's factor the quadratic equation:

2(2y^2 + 3y - 5) > 0

Next, let's solve the equation (2y^2 + 3y - 5 = 0) to find the critical points:

Using the quadratic formula: y = (-b ± √(b^2 - 4ac)) / (2a) a = 2, b = 3, c = -5

y = (-3 ± √(3^2 - 4 × 2 × -5)) / (2 × 2)

y = (-3 ± √(9 + 40)) / 4

y = (-3 ± √49) / 4

y = (-3 ± 7) / 4

The solutions for y are:

y₁ = (-3 + 7) / 4 = 4 / 4 = 1 y₂ = (-3 - 7) / 4 = -10 / 4 = -2.5

Now, let's analyze the inequality using these critical points:

For y < -2.5, the quadratic term (2y^2 + 3y - 5) > 0

For -2.5 < y < 1, the quadratic term (2y^2 + 3y - 5) < 0

For y > 1, the quadratic term (2y^2 + 3y - 5) > 0

Finally, let's consider the sign of the coefficient 2:

When 2y^2 + 3y - 5 > 0, the coefficient 2 is positive.

When 2y^2 + 3y - 5 < 0, the coefficient 2 is negative.

Based on this information, we can conclude:

The solution to the inequality (y × 5) (y + 2) - (y + 2)² > 6 is:

y < -2.5 or y > 1