Problem 36 - Algebra for College Students

Question

Solve each equation. $$5 x^{4}-32 x^{2}+48=0$$

Step-by-Step Solution

Answer

The solutions are \( x = 2, -2, \approx 1.55, \approx -1.55 \).

Step 1: Identify Substitution Variable

Recognize that the equation can be simplified by substituting \( y = x^2 \). This transforms the original equation into a quadratic form.

Step 2: Substitute and Simplify

After substituting \( y = x^2 \), the equation becomes \( 5y^2 - 32y + 48 = 0 \).

Step 3: Solve the Quadratic Equation

Use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 5 \), \( b = -32 \), and \( c = 48 \). Calculate the discriminant \( b^2 - 4ac = (-32)^2 - 4(5)(48) \).

Step 4: Calculate Discriminant

Compute the discriminant: \( 1024 - 960 = 64 \). Since the discriminant is positive, there are two real solutions.

Step 5: Find Roots of Quadratic

Calculate the roots: \( y = \frac{32 \pm \sqrt{64}}{10} \). This simplifies to \( y = \frac{32 \pm 8}{10} \), giving roots \( y = 4 \) and \( y = 2.4 \).

Step 6: Reverse Substitution

Replace \( y \) back with \( x^2 \): Solve \( x^2 = 4 \) and \( x^2 = 2.4 \).

Step 7: Solve for x

For \( x^2 = 4 \), \( x = \pm 2 \). For \( x^2 = 2.4 \), \( x = \pm \sqrt{2.4} \).

Step 8: Simplify Square Roots

Approximate \( x = \pm \sqrt{2.4} \) to two decimal places: \( x \approx \pm 1.55 \).

Step 9: List All Solutions

Combine the solutions from both cases: \( x = 2, -2, \approx 1.55, \approx -1.55 \).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). The equation describes a parabola on a graph. In the context of the given exercise, after substituting \( y = x^2 \), the polynomial is converted into a quadratic form as \( 5y^2 - 32y + 48 = 0 \). This transformation makes it easier to solve since quadratic equations have well-established methods, such as factoring, completing the square, and using the quadratic formula. In our exercise, the last method is preferred due to its direct and systematic approach.

Discriminant Calculation

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is an expression that determines the nature and number of roots the equation possesses. It is given by \( b^2 - 4ac \). In our exercise, for the transformed equation \( 5y^2 - 32y + 48 = 0 \), we calculate the discriminant as:

• \( b = -32 \)
• \( a = 5 \)
• \( c = 48 \)

Thus, the discriminant computation becomes \( (-32)^2 - 4(5)(48) = 1024 - 960 = 64 \). A positive discriminant, like 64, indicates two distinct real roots. A zero discriminant suggests one real root, and a negative one points to complex roots.

Substitution Method

The substitution method can simplify solving higher-degree polynomial equations. In cases where trigonometric or power expressions create complex equations, substituting variables reduces these complexities. In this exercise, the original equation \( 5x^4 - 32x^2 + 48 = 0 \) is solved by substituting \( y = x^2 \). This transforms it into a quadratic equation that is easier to handle. Once the solutions for \( y \) are found, substitution reversal is done to find the values of \( x \). This approach not only simplifies solving the equation but also conveniently applies well-known quadratic solving techniques.

Roots of Equations

The roots of an equation are the values of the variable that satisfy the equation. For quadratic equations, these can be found using the quadratic formula: \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). In our problem, after determining the discriminant, we calculate the roots \( y = \frac{32 \pm \sqrt{64}}{10} \), giving \( y = 4 \) and \( y = 2.4 \). After reversing the substitution by setting \( x^2 = y \), we solve:

• For \( x^2 = 4 \), \( x = \pm 2 \)
• For \( x^2 = 2.4 \), \( x = \pm 1.55 \) approximately

Thus, the roots for the original polynomial equation are \( x = 2, -2, 1.55, \text{and} -1.55 \). Each root is an intersection point with the x-axis on the graphical parabola representation.