Question

solve by making a u-substitution.
\\x^4 - 5x^2 + 4 = 0\\
select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the solution set is \\{\quad\\}
(simplify your answer. use a comma to separate answers as needed.)

b. there is no solution.

Explanation:

🆕 New Concept Discovered: Equations Quadratic in Form
Using substitution to simplify higher-degree equations.

Step 1: Apply u-substitution

We want to solve the equation:

$$ x^4 - 5x^2 + 4 = 0 $$

Notice that the exponent of the first term, \( x^4 \), is the square of the exponent of the middle term, \( x^2 \). This means we can treat it like a quadratic equation by substituting a new variable.

Let:

$$ u = x^2 $$

Squaring both sides gives:

$$ u^2 = (x^2)^2 = x^4 $$

Substituting these into our original equation yields a standard quadratic equation:

$$ u^2 - 5u + 4 = 0 $$

Step 2: Solve the quadratic equation for u

Now we solve for \( u \) by factoring. We look for two numbers that multiply to \( 4 \) and add up to \( -5 \). These numbers are \( -1 \) and \( -4 \).

$$ (u - 1)(u - 4) = 0 $$

Set each factor to zero:

$$ u - 1 = 0 \implies u = 1 $$

$$ u - 4 = 0 \implies u = 4 $$

Step 3: Substitute back to solve for x

Since our original variable is \( x \), we must substitute \( x^2 \) back in place of \( u \).

For \( u = 1 \):

$$ x^2 = 1 $$

$$ x = \pm\sqrt{1} $$

$$ x = 1, -1 $$

For \( u = 4 \):

$$ x^2 = 4 $$

$$ x = \pm\sqrt{4} $$

$$ x = 2, -2 $$

Combining all solutions gives the set: \( \{-2, -1, 1, 2\} \).

Answer:

A. The solution set is { -2, -1, 1, 2 }