How many real zeros please help!!!!!!!!!!!!!!!!

Question

does y=-2(x+3)^3+8?

Answer 1

given equation is 

y=-2(x+3)^3+8----------1

let  x =0   , substitute the x =0 in the equation  1

    y = -2 ( 0+ 3 )^3 + 8

   y = -2 ( 3 )^3 +8

  y = - 54 + 8

 y = -46

 ( x , y ) = ( 0 , -46)

let  x =1 , substitute the x =1 in the equation  1

    y = -2 ( 1+ 3 )^3 + 8

   y = -2 ( 4)^3 +8

  y = - 128 + 8

 y = -120

 ( x , y ) = ( 1 , -120 )

let  x =2 , substitute the x =2 in the equation  1

    y = -2 ( 2+ 3 )^3 + 8

   y = -2 ( 5 )^3 +8

  y = - 250+ 8

 y = -242

 ( x , y ) = ( 2 , -242 )

so the solution ( x , y) ={ ( 0 ,-46 ), ( 1 ,120), (2 ,-242) ...........}

 

 

 

 

 

 

 

Answer 2

The equation is y = -2(x + 3)³ + 8.

y = -2(x + 3)³ + 8

y = -2( x³ + 3³ + 3x²(3) + 3x(3²) ) + 8   [Since (a+b)³ = a³ + b³ +3a²b + 3ab²]

y = -2( x³ + 27 + 9x² + 3x(9) ) + 8

y = -2( x³ + 27 + 9x² + 27x ) + 8

y = -2( x³ ) - 2(27) -2 (9x² ) -2(27x ) + 8

y = -2( x³ ) - 2(27) -2 (9x² ) -2(27x ) + 8

y = -2x³ - 54 -18x² - 54x + 8

y = -2x³ -18x² - 54x + 8 - 54

y = -2x³ -18x² - 54x - 46

The nature of the roots

Every cubic equation with real coefficients has at least one solution x among the real numbers.

This is a consequence of the Intermediate value theorem.

We can distinguish several possible cases using the discriminant,

The following cases need to be considered:

• If  Δ > 0, then the equation has three distinct real roots.
• If  Δ = 0, then the equation has a multiple root and all its roots are real.
• If  Δ < 0, then the equation has one real root and two nonreal complex conjugate roots.

From the equation  -2x³ -18x² - 54x - 46 = 0, a = -2, b = -18, c = -54 and d = -46.

∴ ∆ = 18(-2)(-18)(-54)(-46) - 4(-18)³(-46) + (-18)² (-54)² - 4(-2)(-54)³ - 27(-2)²(-46)²

= 1609632 - 4(-5832)(-46) + (324)(2916) + 8(-157464) - 27(4)(2116)

= 1609632 - 1073088 + 944784 - 1259712 - 228528

= 2554416 - 2561328

= -6912 < 0

∆ is less than 0, therefore the equation has one real root and two nonreal complex conjugate roots.

Therefore the equation y = -2(x + 3)³ + 8 has one real zero.