Problem 44 - College Algebra

Question

Solve each equation in by making an appropriate substitution. $$ 2 x-7 \sqrt{x}-30=0 $$

Step-by-Step Solution

Answer

\(\sqrt{x} = 5\) and \(\sqrt{x} = -3\), therefore the solutions are \(x = 25\) and \(x = 0\).

Step 1: Make the substitution

Since the equation \(2x - 7\sqrt{x} - 30 = 0\) involves a square root, let's pick an appropriate substitution to simplify the equation. Take \(y = \sqrt{x}\), therefore \(y^2 = x\). Substituting these into the original equation we get \(2y^2 - 7y - 30 = 0\).

Step 2: Solve the quadratic equation

Now, you need to solve the quadratic equation \(2y^2 - 7y - 30 = 0\). You can do that by factoring, completing the square, or using the quadratic formula. Here, factoring can be used: \(2y^2 - 10y + 3y - 30 = 0\) which gives \((2y - 10)(y + 3) = 0\). Thus, \(y = 5\) and \(y = -3\).

Step 3: Substitute back to find x

You then replace \(y\) with \(\sqrt{x}\) in the solutions from Step 2, which gives the final solutions to the exercise as \(x = 25\) and \(x = 0\), taking note that \(x\) cannot be negative in the initial equation.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that has a variable raised to the power of two as its highest degree. It is generally expressed in the standard form as: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants, and \(x\) is the unknown variable. In our exercise, after making a substitution, the problem transformed into the quadratic equation \(2y^2 - 7y - 30 = 0\). This is because we set \(y = \sqrt{x}\), which means \(y^2 = x\), rearranging the problem so that it fits the quadratic form. Quadratic equations can be solved using multiple methods such as factoring, completing the square, or applying the quadratic formula. Understanding and recognizing quadratic equations can be key to solving them efficiently.

Solving Equations

Solving equations involves finding the value of the unknown variable that satisfies the equation. The aim is to isolate the variable on one side while maintaining balance within the equation on both sides. **Given the equation**: \[ 2x - 7\sqrt{x} - 30 = 0 \] the task is to find the value of \(x\) that makes the equation true. Instead of solving directly due to its complexity, a substitution was used. This technique is particularly handy in problems that contain radicals or higher powers. Making the equation simpler is the first step to an easier solution.

Factoring

Factoring is a method used to solve quadratic equations, where the equation is rewritten as a product of two binomials. In the problem, the equation \(2y^2 - 7y - 30 = 0\) was factored into \[ (2y - 10)(y + 3) = 0 \] Breaking down these factors can help find the roots of the equation. Each factor is separately set to zero, resulting in two potential solutions. In this case, \(2y - 10 = 0\) gives \(y = 5\), and \(y + 3 = 0\) gives \(y = -3\). Factoring simplifies the process of solving the quadratic equation by revealing its solutions as zeros of the given factors.

Square Root Substitution

Square root substitution is a clever technique often used when equations involve square roots, like the exercise presented. Initially, the term \(\sqrt{x}\) made it challenging to solve directly. By setting \(y = \sqrt{x}\) and substituting \(y^2\) for \(x\), the problem simplifies to a form that is much easier to handle. This removes the square root from the equation, allowing it to be addressed as a standard quadratic. Once solved, substitute back to find the value of the original variable. For example, \(y = 5\) leads to \(x = 25\), confirming the connection between the substitution form and the original variable. This method alleviates complexities and reveals straightforward solutions by transforming the equation format.