Question 1

What does it mean for an equation to be in quadratic form?

Accepted Answer

It can be written as au² + bu + c = 0 for some expression u. The clue is that one term's variable part is the square of another's — x⁴ is the square of x², so x⁴ - 5x² + 4 = 0 is quadratic in u = x².

Question 2

How do I choose u?

Accepted Answer

Let u be the expression whose square appears in the highest-degree term. For x⁴, take u = x²; for x^(2/3), take u = x^(1/3); for (x + 1)², take u = x + 1.

Question 3

Do I always get four solutions from a quartic?

Accepted Answer

Not always. Each u-value gives up to two x-values from u = x². A negative u gives complex x; a repeated u gives fewer distinct x.

Question 4

When do I check for extraneous solutions?

Accepted Answer

Whenever the substitution or back-substitution uses an even root — like u = √(x), which cannot be negative. Reject any u-value that violates that.

Question 5

What is the last step?

Accepted Answer

Back-substitute: replace u with the original expression and solve for the original variable. Solving for u is never the final answer.

Question 6

How do I enter four solutions at once?

Accepted Answer

List them separated by commas in any order. The ± shorthand saves typing: ±1, ±2 is read as the four values -2, -1, 1, 2. Where a substitution produces roots that are not real, give only the real ones unless the problem asks otherwise.

Equations Reducible to Quadratic Form Practice Problems