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Chapter 3: Problem 45

Find all real numbers \(x\) that satisfy the indicated equation. $$ x^{4}-3 x^{2}=10 $$

Short Answer

Expert verified
The real number values for x that satisfy the given equation are \(x = \pm\sqrt{5}\).

Step by step solution

01

Rewrite the equation in terms of \(x^2\)

Let \(y = x^2\). We can now rewrite the given equation in terms of y: $$ y^2 - 3y = 10 $$
02

Solve the quadratic equation for y

Now, we have to solve this quadratic equation for y. First, move the constant term to the other side: $$ y^2 - 3y - 10 = 0 $$ Next, we will attempt to factor the quadratic equation: $$ (y - 5)(y + 2) = 0 $$ From the equation, we have two possible values for y: $$ y - 5 = 0 \Rightarrow y = 5 \\ y + 2 = 0 \Rightarrow y = -2 $$
03

Find the values for x from the given y values

Now that we have the values for y, we can find the values for x by substituting y back with \(x^2\) For \(y = 5\): $$ x^2 = 5 \\ x = \pm\sqrt{5} $$ For \(y = -2\): Since the square of any real number is positive, it is impossible to have \(x^2 = -2\). Therefore, there are no real solutions for this case.
04

Write the final solution

The real number values for x that satisfy the given equation are: $$ x = \pm\sqrt{5} $$

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