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Evaluate a polynomial using a given quadratic equation

Look for the expression already fixed by the equation

A given equation often fixes a combination of terms even when the variable itself has more than one possible value. Rewrite the target expression in terms of that combination before solving for the variable.

Problem

Given $x^{2} - x - 1 = 0$ , find $C = x^{4} - 2 x^{3} + x^{2}$ .

Answer

C = 1

The same value is obtained for either root of the given equation.

Step-by-step solution

1. Read the useful relation

The given equation implies $x^{2} - x = 1$ .

2. Rewrite the target as a square

Factor out $x^{2}$ and recognize the remaining perfect square:

C = x^{2} (x^{2} - 2 x + 1) = x^{2} (x - 1)^{2} = (x^{2} - x)^{2} .

3. Substitute the known combination

Now $C = 1^{2} = 1$ . There is no need to calculate the individual roots of the quadratic.

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