Chapter 1, Problem 78RE

To determine

To calculate: The real solutions of the equation $x^4-x^2-9=0$ .

Answer to Problem 78RE

The solutions of the equation are $x=1.594$ and $x=-1.594$ .

Explanation of Solution

Given information:

The equation is given as $x^4-x^2-9=0$ .

Formula used:

In order to find all the solutions to higher-degree equation, use synthetic division, factoring, and the Quadratic Formula.

In order to Factorise the high degree polynomial, determine all the terms that were multiplied together to get the given polynomial. Then try to factor each of the terms found in the first step. This continues until it can’t be factored anymore. When it can’t be factored, then polynomial is completely factored.

For an equation of the form $a{x}^2+bx+c=0$ , the solution is given by quadratic formula given by:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

Calculation:

Consider the equation $x^4-x^2-9=0$ .

Convert this equation into quadratic equation by taking $x^2=w$ .

Then, the equation $x^4-x^2-9=0$ can be written as $w^2-w-9=0$ .

Therefore, the solution of this equation is given by $w=\frac{-(-1)\pm \sqrt{{(-1)}^2-4(1)(-9)}}{2(1)}=\frac{1\pm \sqrt{1+36}}{2}=\frac{1\pm \sqrt{37}}{2}=2.54$ .

Now , since $x^2=w$ . Hence, $x=\pm \sqrt{2.54}$

Thus, the real solutions of $x^4-x^2-9=0$ are $x=1.594$ and $x=-1.594$ .