Solving equations

1. Solve the equation4x^3－8x+4=0
Sol
4x^3－8x+4=0
x^3－2x+1=0
(x^2+x－1)(x－1)=0
x=(－1+/-√5)/2 or x=1

2. y= 2x^3－3x^2+3. Find x-intercepts.
Sol
2x^3－3x^2+3=0
2(x^3+0.8055818x^2)－4.6111636(x^2+0.8055818x)+3.7146694(x+0.8055818)=0
(2x^2－4.6111636x^2+3.7146694)(x+0.8055818)=0
4,.6111636^2－4*2*3.7146694=－8.454526<0
x=－0.8055818

3. y=－x^3+27x－36. Find x-intercepts
Sol
－x^3+27x－36=0
x^3－27x+36=0
(x^3+5.765353x^2)－5.765353(x^2+5.765353x)+6.239295(x+5.765353)=0
(x^2－5.765353x+6.239295)(x+5.765353)=0
x=[5.765353+/-√(5.765353^2－4*1*6.239295)]/2 or x=－5.765353
x=4.3216096or 1.4437277 or －5.765353

4. y= x^3+2x^2－x+1. Find x-intercepts.
Solx^3+2x^2－x+1=0
(x^3+2.546558x^2)－0.546558(x^2+2.546558x)+0.3918416(x+2.546558)=0
(x^2－0.546558x+0.3918416)(x+2.546558)=0
0.546558^2－4*1*2.546558=－9.887507<0
x=－2.546558

5. y=－2x^3+2x+5. Find x-intercepts.
Sol
－2x^3+2x+5=0
2x^3－2x－5=0
2(x^3－1.60143x^2)+3.20286(x^2－1.60143x)+3.129156(x－1.60143)=0
(2x^2+3.20286x+3.129156)(x－1.60143)=0
3.20286^2－4*2*3.129156=－14.774936<0
x=1.60143

Using Cardano formula (accurate to 6 decimals)

2: The x-intercept: x = - 0.8064439

3: The x-intercept: x = -.5.765741, x = 1.445106, x = 4.320636

4: The x-intercept: x = -.2.546818

5: The x-intercept: x = 1.600599