Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the domain of \(x\) in the expression. $$\sqrt[4]{2 x^{2}+4 x+3}$$

Step-by-Step Solution

Answer

The domain of \(x\) in the expression \(\sqrt[4]{2 x^{2}+4 x+3}\) is all real numbers, denoted as (-∞, ∞)

Step 1: Frame the inequality

To find the domain of \(x\), form an inequality that states the expression inside the root should be greater than or equal to 0. \(2x^{2} + 4x + 3 \geq 0\)

Step 2: Solve the inequality

Solving this inequality will give us the set of \(x\) values that make the entire expression real. It's a quadratic inequality, so we first find the roots of the equivalent equation \(2x^{2} + 4x + 3 = 0\)

Step 3: Find the roots

We can solve for the roots of the equivalent equation by first dividing all terms by 2 to simplify, resulting in \(x^{2} + 2x + 3/2 = 0\). Then subtract 1 from both sides of the equation to complete the square on the left side, resulting in \(x^{2} + 2x + 1 = -1/2\). Therefore, \((x+1)^2 = -1/2\). Here, we see that the square of a real number is negative, which is impossible. Therefore, the equation has no real roots.

Step 4: Conclude the inequality solution

The inequality \(2x^{2} + 4x + 3 \geq 0\) is therefore always true for all real numbers, since the quadratic function opens upwards and it does not intersect the x-axis (has no real roots).

Key Concepts

Solving Quadratic Inequalities

Understanding how to solve quadratic inequalities is essential when determining the domain of certain functions. Much like the process of solving quadratic equations, you begin by moving all terms to one side of the inequality to set it to zero. The standard form of a quadratic inequality is \( ax^2 + bx + c \geq 0 \), though it could also be \( \leq 0 \) depending on the context.

Once in standard form, the next step is to identify the roots of the associated quadratic equation \( ax^2 + bx + c = 0 \). This is often achieved by factoring, completing the square, or using the quadratic formula. However, unlike equations, solving an inequality does not just involve finding the exact values that satisfy the equality. Once we have the roots, we sketch the quadratic's graph to visualize the intervals on which the function is positive (above the x-axis) or negative (below the x-axis), keeping in mind the shape of a parabola. After determining these intervals, we conclude which satisfy the original inequality and thus, the solution set.

Quadratic Function Domain

The domain of a function refers to all the possible input values (usually x-values) that the function can accept to produce real number outputs. For a quadratic function, the general form is \( y = ax^2 + bx + c \), and its graph is a parabola. The domain of a standard quadratic function is all real numbers, \( -\infty < x < \infty \), because there are no x-values that make the function undefined.

However, when we deal with the domain of expressions that involve a square root of a quadratic function, like \( \sqrt[4]{2x^2 + 4x + 3} \), we must ensure that the value under the radical is non-negative to belong in the real number system. That's because you cannot take an even root of a negative number and get a real result. Therefore, we set the inside of the radical greater than or equal to zero and solve the resulting inequality to find the domain, which might not be all real numbers in this case.

Analyzing Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. When analyzing radical expressions, especially those with an even-indexed root (like a square root or fourth root), it is crucial to consider the expression within the radical to be non-negative, as the even roots of negative numbers are not defined within the real number system.

The expression \( \sqrt[4]{2x^2 + 4x + 3} \) has a fourth root, which is an even-indexed root, meaning we need to make sure that the value of \( 2x^2 + 4x + 3 \) is \( \geq 0 \). If the expression inside the radical were negative, the fourth root would not yield a real number, which is why we solve the inequality \( 2x^2 + 4x + 3 \geq 0 \) to determine the domain. The process often involves finding the roots of the underlying quadratic equation, which informs us about the intervals where the quadratic expression maintains a non-negative value. Although in this example the inequality already holds true for all real numbers, this approach is key when handling more complex radical expressions.