Problem 116 - Introductory Algebra for College Students

Question

Factor completely. $$7 x^{4}+34 x^{2}-5$$

Step-by-Step Solution

Answer

The completely factored form of the expression \(7 x^{4}+34 x^{2}-5 = 0\) is \((x^2-0.2)(x^2+3.6) = 0\).

Step 1: Substitution

Instead of dealing with \(7x^{4}+34x^{2}-5\) directly, substitute \(y\) for \(x^2\). The expression becomes \(7y^2 + 34y - 5\). Now it's recognizable as a quadratic equation.

Step 2: Factoring Quadratic Equation Using Formula

Factorize the quadratic equation. The standard form of a quadratic equation is \(ay^2 + by + c = 0\), where \(a=7\), \(b=34\), and \(c=-5\). The formula for factoring a quadratic equation is \((y-p)(y-q)=0\), where \(p\) and \(q\) are the roots of the equation. Calculate the roots using the quadratic formula \(p, q = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\). The roots obtained are \(p=0.2\) and \(q=-3.6\). Therefore, the factored form of the quadratic equation is \((y-0.2)(y+3.6) = 0\).

Step 3: Substituting Back

Go back to the original equation by substituting \(y\) back with \(x^2\). The completely factored expression becomes: \((x^2-0.2)(x^2+3.6) = 0\).