PLEASE HELP. Solve these equations, please?

Question

I'm stuck on these problems. I know the answer. Just not how to get there. Please help!

1.) x^3 + 4x^2 - 3x - 2 = 0

2.) 16x^4 - 4x^3 - 16x^2 + x + 3 = 0

3.) 3(square root ->3x+14) = 15 - 2 | x - 2 |

WILL GIVE BEST ANSWER.

Answer 1

1.) x^3 + 4x^2 - 3x - 2 = 0
Hint:
1+4-3-2=0
is true.
Find one solution and then solve the remaining quadratic equation.
This is a similar problem like the second one, but much easier.
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2.) 16x^4 - 4x^3 - 16x^2 + x + 3 = 0

It is an equation of 4th degree and generally it is not easy to solve. Some tricks have to be used.

First trick:
We see that
16 – 4 – 16 + 1 + 3 = 0
is true.
Therefore the first solution is
x = 1.

Divide the equation by (x-1).
(x-1)*(16x^3+12x^2-4x-3)=0
Now we have to solve cubic equation
16x^3 + 12x^2 - 4x - 3 = 0
16x^3 - 4x + 12x^2 - 3 = 0
4 x (4x^2 – 1) + 3 ( 4x^2 – 1) = 0
(4x^2 – 1) * (4x + 3) = 0
The remaining three solutions are
x = 0.5
x = -0.5
and
x = -0.75
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3.) 3(square root ->3x+14) = 15 - 2 | x - 2 |

3*((3*x+14)^0.5) = 15 – 2*|x-2|

First let us find solutions x ≤ 2:

3*((3*x+14)^0.5) = 15 + 2*(x-2)
3*((3*x+14)^0.5) = 2*x + 11
9*(3*x+14) = 4*x^2 + 44*x + 121
27*x+126 = 4*x^2 + 44*x + 121
4*x^2 + 17*x – 5 = 0

Solutions are
x=0,276171589
and
x=-4,526171589.

Second let us find solutions x > 2:

3*((3*x+14)^0.5) = 15 - 2*(x-2)
3*((3*x+14)^0.5) = 15 - 2*x +4
3*((3*x+14)^0.5) = -2*x +19
9*(3*x+14) = 4*x^2 – 76*x + 361
27*x+126 = 4*x^2 – 76*x + 361
4*x^2 – 103*x + 235 = 0

Solutions are
x=23,21983567
and
x=2,530164332.

There are four solutions altogether.
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