Problem 44 - College Algebra Essentials

Question

By making an appropriate substitution. $$ 4 x^{4}=13 x^{2}-9 $$

Step-by-Step Solution

Answer

The solutions to the equation are x = \( \sqrt{1.5} \) , and - \( \sqrt{1.5} \).

Step 1: Substitution

Replace \(x^{2}\) with \(t\). The equation therefore transforms to \(4t^{2} - 13t + 9 = 0\).

Step 2: Solving Quadratic Equation

The equation is now a quadratic equation, which can be solved by using the quadractic formula \(-b \pm \sqrt{b^{2} - 4ac} / 2a\), here a=4, b=-13 and c=9. Substituting these values into the formula, the solutions for \(t\) become \(t1 = 1.5 , t2 = 1.5\).

Step 3: Reversing Substitution

Now, substitute \(t\) back into the equation with the original variable, \(x\), this yields the solutions for \(x\) as \(x=\sqrt{1.5} , x=-\sqrt{1.5}\).

Key Concepts

Substitution Method

The substitution method is a handy technique often used to simplify equations, especially when dealing with polynomial equations that would otherwise be complex to solve. It involves replacing a part of the equation with another variable to make the calculation more manageable. Let's examine how this works with the provided equation: \(4x^4 = 13x^2 - 9\).

• First, identify a substitution variable. Here, by letting \(x^2 = t\), the equation simplifies.
• Substitution transforms the equation into a simpler, more familiar form, like a quadratic: \(4t^2 - 13t + 9 = 0\).

Recognizing when and how to use substitution can significantly streamline solving complex equations like this one.This method simplifies the problem temporarily, allowing use of familiar solving techniques. After finding solutions in terms of \(t\), reverse the substitution to return to the original variable, \(x\). This strategy is particularly useful when encountering higher-degree polynomial equations that contain squared terms.

Quadratic Formula

The Quadratic Formula is a crucial tool for solving quadratic equations, which take the general form \(ax^2 + bx + c = 0\). This formula provides a direct way to find the roots (solutions) of the equation by plugging in the coefficients \(a\), \(b\), and \(c\).

• The formula is \(-b \pm \sqrt{b^2 - 4ac} / 2a\).
• This method guarantees finding the roots for any quadratic equation as long as \(a eq 0\).
• In our transformed equation \(4t^2 - 13t + 9 = 0\), we applied the formula by setting \(a = 4\), \(b = -13\), and \(c = 9\).

By substituting these values into the Quadratic Formula, we derived the roots \(t_1 = 1.5\) and \(t_2 = 1.5\), both of which are valid because the discriminant \(b^2 - 4ac\) is zero. This shows that the quadratic has a double root, meaning both roots are equal, a scenario possible when the parabola (graph of the quadratic) touches the x-axis at one point.

Polynomial Equations

Polynomial equations are equations formed by polynomials, which are expressions consisting of variables and coefficients combined through addition, subtraction, multiplication, and non-negative integer exponents. The degree of a polynomial is the highest power of the variable in the expression.

• In the original exercise, the given polynomial equation is a quartic equation (degree 4): \(4x^4 = 13x^2 - 9\).
• Understanding polynomials includes recognizing their standard form and degree, which helps in determining the appropriate methods for solving.
• Substitution converted this quartic polynomial into a quadratic form, which is easier to solve.

This manipulation shows the power of algebraic techniques like substitution to tackle higher-degree polynomials by temporarily reducing them to lower-degree models that are simpler to manage. Polynomial equations appear in various mathematical contexts, and mastering methods to solve them effectively is essential for deeper mathematical problem-solving.