Question

Solve the equation x^4+3x-2=0

Answer 1

The solutions to x4 - 3x2 + 2 = 0 are x = -1, 1, -√(2), and √(2).

To solve this equation, we will use substitution. That is, we will let m = x2, then plug it in anywhere we see x2 in the equation. This gives the following equation in m.

m2 - 3m + 2 = 0

Now, we can solve this equation for m, then plug our original substitution back in, and then solve for x. This whole process is as follows:

x4 - 3x2 + 2 = 0

Let m = x2, and plug m in for x2 in this equation.

m2 - 3m + 2 = 0

Factor the left-hand side of the equation.

(m - 1)(m - 2) = 0

Set each factor equation to zero to get m - 1 = 0 and m - 2 = 0. Solve each of these equations for m.

m - 1 = 0

Add 1 to both sides of the equation.

m = 1

m - 2 = 0

Add 2 to both sides of the equation.

m = 2

Plug x2 back in for m into each of these equations to get x2 = 1 and x2 = 2. Solve both of these equations for x.

x2 = 1

Take the square root of both sides of the equation, and add in the ± symbol on the right-hand side.

x = ± √(1) = ±1

x2 = 2

Take the square root of both sides of the equation, and add in the ± symbol on the right-hand side.

x = ± √(2)

All together, we get that the solutions to the equation x4 - 3x2 + 2 = 0 are x = -1, 1, -√(2), and √(2).

Answer 2

Correct Question: Solve the equation $x^4-3x^2+2=0$.

Answer:

The four roots of the equation $x^4-3x^2+2=0$ are -1, 1, $-\sqrt{2}$ and $\sqrt{2}$.

Step-by-step explanation:

Consider the given equation as follows:

$x^4-3x^2+2=0$ _____ (1)

Let $x^2=t$. Then,

Equation (1) becomes,

$t^2-3t+2=0$

Notice that equation (1) turns into the quadratic equation of one variable.

Using the middle-term splitting method, simplify as follows:

$t^2-2t-t+2=0$

$t(t-2)-1(t-2)=0$

$(t-1)(t-2)=0$

t = 1 and t = 2

Case1. When t = 1. Then,

$x^2=1$ (Since $x^2=t$)

Take the square root on both the sides, we get

$x=\pm\sqrt{1}$

$x=\pm1$

x = -1 and x = 1

Case2. When t = 2. Then,

$x^2=2$ (Since $x^2=t$)

Take the square root on both the sides, we get

$x=\pm\sqrt{2}$

$x = -\sqrt{2}$ and $x = \sqrt{2}$

Therefore, the four roots of the equation $x^4-3x^2+2=0$ are -1, 1, $-\sqrt{2}$ and $\sqrt{2}$.

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