Problem 75 - Algebra for College Students

Question

Solve each equation. $$x-9 \sqrt{x}+18=0$$

Step-by-Step Solution

Answer

The solutions are \( x = 36 \) and \( x = 9 \).

Step 1: Make a substitution

Let \( y = \sqrt{x} \), which implies \( x = y^2 \). Replace \( x \) in the equation \( x - 9\sqrt{x} + 18 = 0 \) with \( y^2 - 9y + 18 = 0 \).

Step 2: Identify the quadratic equation

Now, we have a quadratic equation in terms of \( y \): \( y^2 - 9y + 18 = 0 \). This equation can be solved using the quadratic formula.

Step 3: Apply the quadratic formula

The quadratic formula is \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -9 \), and \( c = 18 \). Substitute these values into the formula: \( y = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot 18}}{2\cdot 1} \).

Step 4: Simplify the quadratic formula

Calculate \( y = \frac{9 \pm \sqrt{81 - 72}}{2} \). Simplify the square root and perform the operations: \( y = \frac{9 \pm \sqrt{9}}{2} \), which is \( y = \frac{9 \pm 3}{2} \).

Step 5: Find the values of y

Solving for \( y \), we have two potential solutions: \( y = \frac{9 + 3}{2} = 6 \) and \( y = \frac{9 - 3}{2} = 3 \).

Step 6: Relate back to x

Recall \( y = \sqrt{x} \). Thus, \( \sqrt{x} = 6 \) gives \( x = 36 \), and \( \sqrt{x} = 3 \) gives \( x = 9 \).

Step 7: Verify the solutions

Substitute \( x = 36 \) and \( x = 9 \) back into the original equation to check. For \( x = 36 \), the equation holds: \( 36 - 9\times6 + 18 = 0 \). Also, for \( x = 9 \), the equation holds: \( 9 - 9\times3 + 18 = 0 \). Both satisfy the original equation.

Key Concepts

Substitution Method

The substitution method is a handy technique used to simplify equations, making them easier to solve. In the given problem, we encounter the equation \(x - 9\sqrt{x} + 18 = 0\). With square roots involved, this equation may appear complex at first glance. To address this, we use substitution.By setting \(y = \sqrt{x}\), we transform the equation from being in terms of \(x\) to being in terms of \(y\). This results in a simpler quadratic form: \(y^2 - 9y + 18 = 0\). Now, the problem can be approached with methods suitable for quadratic equations, such as factoring or using the quadratic formula. The substitution helps us by:

• Changing the structure of the equation to a more familiar form.
• Allowing the use of established methods to find solutions.

In this problem, the substitution method transforms a situation involving square roots into a basic quadratic equation, ready for the next steps in solving.

Quadratic Formula

Once the quadratic equation \(y^2 - 9y + 18 = 0\) is established through substitution, we employ the quadratic formula to find its solutions. The quadratic formula is a powerful tool used to solve any quadratic equation of the form \(ax^2 + bx + c = 0\). It is expressed as:\[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For our equation:

• \(a = 1\)
• \(b = -9\)
• \(c = 18\)

Substituting these values into the quadratic formula gives us:\[y = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot 18}}{2 \cdot 1}\]Simplifying under the square root and performing the calculations, we find the two solutions: \(y = 6\) and \(y = 3\). These represent the values of \(y\) that satisfy the equation \(y^2 - 9y + 18 = 0\).This formula is especially useful since it guarantees finding all possible solutions for any quadratic equation, provided you correctly substitute the parameters \(a\), \(b\), and \(c\). In our case, it neatly delivers the values we need.

Verifying Solutions

Verifying solutions is an essential step in solving mathematical equations to ensure the accuracy of the anticipated results. After finding the values \(y = 6\) and \(y = 3\) through the quadratic formula, it's crucial to check whether they satisfy the original equation.Substituting back, we first convert these \(y\) values into \(x\) values:

• If \(y = 6\), then \(\sqrt{x} = 6\), leading to \(x = 36\).
• If \(y = 3\), then \(\sqrt{x} = 3\), leading to \(x = 9\).

With these \(x\) values, we substitute each back into the original equation \(x - 9\sqrt{x} + 18 = 0\):- For \(x = 36\): - Calculate: \(36 - 9 \times 6 + 18 = 0\) confirms solution validity.- For \(x = 9\): - Calculate: \(9 - 9 \times 3 + 18 = 0\) also confirms validity.This step checks whether the solutions truly solve the initial equation and reassures us that no errors took place during substitution or calculation. It's always a good routine to verify your work as mathematical content can sometimes be complex, and procedural mistakes should be caught and corrected.