Problem 77 - Algebra for College Students

Question

Solve each equation. $$x+\sqrt{x}-2=0$$

Step-by-Step Solution

Answer

The solution is \(x = 1\).

Step 1: Isolate the square root

Start by moving the constant term over to the other side of the equation to isolate the square root term: \( x + \sqrt{x} = 2 \). Subtract \(x\) from both sides to get \( \sqrt{x} = 2 - x \).

Step 2: Square both sides of the equation

Square both sides of the equation to eliminate the square root: \( (\sqrt{x})^2 = (2 - x)^2 \). This simplifies to \( x = 4 - 4x + x^2 \).

Step 3: Rearrange the equation

Rearrange the equation to form a quadratic equation: \( x^2 - 5x + 4 = 0 \).

Step 4: Factor the quadratic equation

Find two numbers that multiply to 4 and add to -5. These numbers are -1 and -4. So, factor the quadratic equation: \( (x-1)(x-4) = 0 \).

Step 5: Solve for x

Set each factor equal to zero and solve for \( x \): \( x - 1 = 0 \) gives \( x = 1 \), and \( x - 4 = 0 \) gives \( x = 4 \).

Step 6: Verify solutions

Substitute \( x = 1 \) and \( x = 4 \) back into the original equation. For \( x = 1 \), \( 1 + \sqrt{1} - 2 = 0 \) holds true. For \( x = 4 \), \( 4 + \sqrt{4} - 2 = 4 eq 0 \). Only \( x = 1 \) satisfies the original equation.

Key Concepts

Solving Equations

In mathematics, solving equations involves finding the value of the unknown variable that makes the equation true. For the equation \(x + \sqrt{x} - 2 = 0\), our goal is to determine the value of \(x\) such that the left-hand side equals zero. To do this, we start by isolating terms that contain the square root.

• Initially, rearrange terms to get \(x + \sqrt{x} = 2\).
• Further, subtract \(x\) to isolate \(\sqrt{x}\): \(\sqrt{x} = 2 - x\).

Solving an equation can involve several steps such as isolating terms, using algebraic operations, and transforming the equation into a simpler form, like a quadratic equation. This way, we can apply more methods like factoring to find the solutions.

Square Roots

Understanding square roots is crucial when solving equations that include them. A square root is a number which, when multiplied by itself, gives the original number. For instance, \(\sqrt{x}\) represents a number that squares to \(x\). In our equation, the square root appears as \(\sqrt{x}\).
To eliminate the square root from the equation \(\sqrt{x} = 2 - x\), we square both sides:

• Squaring the left side, \( (\sqrt{x})^2 = x\).
• Squaring the right side, \((2 - x)^2 = 4 - 4x + x^2\).

This process transforms our equation into a quadratic form, making it easier to solve.

Factoring

Factoring is a key technique for solving quadratic equations. Once an equation is in the form \(ax^2 + bx + c = 0\), we can factor it to find the roots. In our example, the quadratic equation is \(x^2 - 5x + 4 = 0\). The goal is to express it as a product of two binomials.

• Look for two numbers whose product is the constant term (4) and whose sum is the linear coefficient (-5).
• In this case, the numbers -1 and -4 fit these conditions.

Thus, we can factor the equation as \((x - 1)(x - 4) = 0\). This shows us the potential solutions are \(x = 1\) and \(x = 4\).

Verification of Solutions

Verification is an essential step in solving equations as it confirms the validity of the solutions. After finding possible solutions from the factored form \((x - 1)(x - 4) = 0\), we have \(x = 1\) and \(x = 4\).
To verify, substitute each back into the original equation and check if they satisfy it:

• For \(x = 1\): Substitute in the equation \(1 + \sqrt{1} - 2 = 0\), which holds true.
• For \(x = 4\): Substitute in the equation \(4 + \sqrt{4} - 2 = 4eq 0\), which does not satisfy the equation.

Thus, only \(x = 1\) is a valid solution. Verification ensures that you have the correct and complete answer.