Problem 46 - Beginning and Intermediate Algebra

Question

Solve each equation. Check the solutions. \(p-2 \sqrt{p}=8\)

Step-by-Step Solution

Answer

The solution is \( p = 16 \).

Step 1 - Move terms to one side

Begin by moving all terms in the equation to one side to set the equation to zero: \[ p - 2\sqrt{p} - 8 = 0 \]

Step 2 - Substitute variable

Let \( x = \sqrt{p} \). This substitution means \( p = x^2 \). Substitute \( x \) into the equation: \[ x^2 - 2x - 8 = 0 \]

Step 3 - Solve the quadratic equation

Solve the quadratic equation \( x^2 - 2x - 8 = 0 \) using the quadratic formula. The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For our equation, \(a = 1\), \(b = -2\), and \(c = -8\): \[ x = \frac{2 \pm \sqrt{4 + 32}}{2} = \frac{2 \pm \sqrt{36}}{2} = \frac{2 \pm 6}{2} \]So, \(x = 4\) or \(x = -2\)

Step 4 - Back-substitute to find \(p\)

Since \( x = \sqrt{p} \), we substitute back to find \( p \):If \(x = 4\), then \( \sqrt{p} = 4 \implies p = 16 \)If \(x = -2\), then \( \sqrt{p} = -2 \). However, this is not possible since the square root of \(p\) cannot be negative. Thus, we discard \(x = -2\)

Step 5 - Check the solution

Check the solution by substituting \( p = 16 \) back into the original equation: \[ p - 2 \sqrt{p} = 8 \] \[ 16 - 2 \times 4 = 8 \]\[ 16 - 8 = 8 \]This is true, so \( p = 16 \) is a valid solution.

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation in the form of ax^2 + bx + c = 0, where a, b, and c are constants, and x represents the variable we need to solve for. In quadratic equations, the highest power of the variable is 2. These equations can be solved using various methods like factoring, completing the square, or using the quadratic formula. The quadratic formula, which is derived from the process of completing the square, is: \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]. In the exercise given, we first rewrite the original equation involving the square root into a quadratic form by making a substitution. This conveniently transformed the problem into solving a standard quadratic equation.

Substitution Method

The substitution method involves replacing a variable or term in an equation with another expression. This helps to simplify complex equations and make them easier to solve. In the given exercise, we used the substitution method by letting \( x = \sqrt{p} \). This allowed us to convert the original equation into a quadratic equation: \[ x^2 - 2x - 8 = 0 \]. We made this substitution based on the identity that \( p = x^2 \). Once we solve for \( x \) using the quadratic formula or other means, we substitute back to find the original variable, \( p \). This manipulation highlights the effectiveness of substitution in simplifying equations and making them more manageable.

Checking Solutions

After solving an equation, it is crucial to verify that the solutions obtained are correct. This process is called checking solutions. For our given exercise, once we determined that \( p = 16 \) by solving the quadratic equation and discarding inconsistent results, we substituted this value back into the original equation: \[ p - 2 \sqrt{p} = 8 \]. Substituting \( p = 16 \) results in: \[ 16 - 2 \times 4 = 8 \]. Since both sides of the equation are equal, the solution \( p = 16 \) is verified as correct. Always make sure to check the solutions to rule out extraneous or incorrect answers, especially in problems involving square roots or other non-linear terms.