Problem 53 - College Algebra

Question

Explain why the equation \(x^{4}+6 x^{2}+2=0\) has no rational roots.

Step-by-Step Solution

Answer

The equation \(x^{4}+6 x^{2}+2=0\) has no rational roots because the discriminant of its associated quadratic equation does not yield a perfect square, implying that the roots of the original equation can't be rational.

Step 1: Rearrange the Equation

Write the equation in the following way: \((x^{2})^{2}+6(x^{2})+2 = 0\). Now it looks like a quadratic equation:\(aX^{2}+bX+c = 0\), where \(X=x^{2}, a=1, b=6, c=2\).

Step 2: Apply the Discriminant

For any quadratic equation, the Discriminant \(D = b^{2}-4ac\) calculates whether the roots are real or imaginary, and rational or irrational. In this case, it's \(D = 6^{2}-4*1*2 = 36-8 =28\). The square root of number 28 is not a rational number.

Step 3: Analyze the Discriminant

Since the discriminant is not a perfect square (not equal to an integer squared), the roots of the equation are irrational when \(X=x^{2}\). Furthermore, since \(x^{2}\) can only be a positive number or zero (for real x), and the roots for \(X\) are irrational and non-zero, there can't be any rational values for x that satisfy the original equation.