Question

Find the real solutions of each equation. $$ 4 x^{1 / 2}-9 x^{1 / 4}+4=0 $$

Short answer

The real solutions are \( x = \left( \frac{9 + \sqrt{17}}{8} \right)^4 \) and \( x = \left( \frac{9 - \sqrt{17}}{8} \right)^4 \).

Step-by-step solution

Substitute a new variable

Let \( y = x^{1/4} \). Then \( y^2 = (x^{1/4})^2 = x^{1/2} \). This transforms the original equation into a quadratic equation: \( 4y^2 - 9y + 4 = 0 \).

Solve the quadratic equation

Use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 4 \), \( b = -9 \), and \( c = 4 \). Calculate the discriminant: \( b^2 - 4ac = (-9)^2 - 4(4)(4) = 81 - 64 = 17 \). Therefore, \( y = \frac{9 \pm \sqrt{17}}{8} \).

Substitute back the original variable

Recall \( y = x^{1/4} \). So, \( x^{1/4} = \frac{9 + \sqrt{17}}{8} \) or \( x^{1/4} = \frac{9 - \sqrt{17}}{8} \). Raise both sides to the 4th power to solve for \( x \): \( x = \left( \frac{9 + \sqrt{17}}{8} \right)^4 \) or \( x = \left( \frac{9 - \sqrt{17}}{8} \right)^4 \).

Verify real solutions

Ensure that both solutions in Step 3 result in real values. Since the discriminant is positive and the transformations are reversible, both \( x = \left( \frac{9 + \sqrt{17}}{8} \right)^4 \) and \( x = \left( \frac{9 - \sqrt{17}}{8} \right)^4 \) lead to valid real numbers.

Key concepts

Substitution in Equations

Substitution is a powerful technique used to simplify complex equations by introducing a new variable. For instance, if we have an equation like \[ 4 x^{1 / 2} - 9 x^{1 / 4} + 4 = 0 \], it’s easier to handle it by letting \[ y = x^{1 / 4} \] and consequently \[ y^2 = x^{1 / 2} \]. This transforms our original equation into a simpler quadratic form, \[ 4y^2 - 9y + 4 = 0 \]. This substitution reduces the complexity and allows us to apply other methods more easily.

Solving Quadratic Equations

A quadratic equation is any equation that can be rewritten in the form of \[ ax^2 + bx + c = 0 \]. Solving such equations involves finding the values of \[ x \] that satisfy this equation. There are several methods to solve quadratic equations, including factorization, completing the square, and using the quadratic formula. In the exercise, the equation \[ 4y^2 - 9y + 4 = 0 \] is a quadratic equation in \[ y \], and we solve it using the quadratic formula. Notice the importance of each coefficient in determining the solution.

Quadratic Formula

The quadratic formula is a universal method to find solutions to any quadratic equation \[ ax^2 + bx + c = 0 \]. It is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. In our example, with \[ a = 4, b = -9, \] and \[ c = 4 \], we substitute these values into the formula to find \[ y \]. So, \[ y = \frac{9 \pm \sqrt{17}}{8} \], giving us two solutions. These values of \[ y \] are further used to retrieve the original variable \[ x \]. The quadratic formula is essential when factorization is not straightforward, making it a fundamental tool for solving quadratic equations.

Discriminant

The discriminant of a quadratic equation, given by \[ b^2 - 4ac \], helps determine the nature of the solutions. It is part of the quadratic formula and tells us whether the solutions are real or complex. In our case, \[ b^2 - 4ac = 17 \], which is positive. This indicates that the solutions are real and distinct. The discriminant is crucial as it quickly informs us of the number and type of solutions without solving the entire equation.