Problem 35 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$x-\sqrt{2 x+3}=0$$

Step-by-Step Solution

Answer

x = 3

Step 1: Isolate the Square Root Term

Move \(-\sqrt{2x + 3}\) to the right side of the equation \(-x + \sqrt{2x + 3} = 0\).

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square root: \(x = \sqrt{2x + 3} \Rightarrow x^2 = 2x + 3\).

Step 3: Rearrange the Equation

Rearrange the equation in the form of a standard quadratic equation: \(x^2 - 2x - 3 = 0\).

Step 4: Factor the Quadratic Equation

Factor the quadratic equation. Find two numbers that multiply to \(-3\) and add to \(-2\): \((x-3)(x+1) = 0\).

Step 5: Solve for x

Set each factor equal to zero and solve for \(x\):\(x-3=0\) and \(x+1=0\). \(x=3\) and \(x=-1\).

Step 6: Check Solutions in Original Equation

Substitute \x = 3\ and \x = -1\ back into the original equation to verify. Only \x=3\ is valid because \(-1\) leads to a non-real solution when substituting back.

Key Concepts

isolating terms

Isolating terms is an essential first step in solving equations. It involves rearranging the equation to get the variable of interest alone on one side. In the context of solving quadratic equations, isolating terms prepares the equation for further manipulation.
In our example, we start with the equation:

• $$x - \sqrt{2x + 3} = 0$$

Our goal is to isolate the square root term on one side. We achieve this by adding \sqrt{2x + 3} to both sides:

• $$x = \sqrt{2x + 3}$$

Isolating terms makes it easier to apply other mathematical techniques, such as squaring both sides, in subsequent steps.

square roots

Square roots are often involved in solving quadratic equations. To eliminate a square root, you can square both sides of the equation. This step is crucial for simplifying equations and making the variable easier to solve.
Here's how we do it in our example:

• Start with the equation after isolating terms: $$x = \sqrt{2x + 3}$$

Then, square both sides to remove the square root:

• $$x^2 = 2x + 3$$

By squaring both sides, we now have a quadratic equation without square roots. This makes it easier to proceed to the next steps, like rearranging and factoring the equation.

factoring quadratics

Factoring quadratics is an important technique for solving quadratic equations. It involves expressing the equation in a product of two binomials. In simpler terms, you break down the quadratic equation into simpler components that multiply to give the original equation.
In our example, after squaring both sides, we have:

• $$x^2 - 2x - 3 = 0$$

The next step is to factor the quadratic equation. We need to find two numbers that multiply to \(-3\) and add to \(-2\):

• $$x^2 - 2x - 3 = (x - 3)(x + 1) = 0$$

Factoring simplifies the equation, breaking it into two linear equations that are easier to solve. This allows us to find the solutions for \( x \) by setting each factor equal to zero.

checking solutions

Checking solutions is a vital step to verify the correctness of your answers. It ensures that the solutions you found satisfy the original equation.
In our example, after factoring, we have two potential solutions:

• $$x - 3 = 0\quad\Rightarrow\quad x = 3$$
• $$x + 1 = 0\quad\Rightarrow\quad x = -1$$

We need to substitute these values back into the original equation to verify them:

• For \( x = 3 \) :
  
  $$ 3 - \sqrt{2(3) + 3} = 0$$
  
  $$ 3 - \sqrt{6 + 3} = 0$$
  
  $$ 3 - \sqrt{9} = 0$$
  
  $$ 3 - 3 = 0 $$ (Valid)

• For \( x = -1 \) :
  
  $$ -1 - \sqrt{2(-1) + 3} = 0$$
  
  $$ -1 - \sqrt{ -2 + 3 } = 0$$
  
  $$ -1 - \sqrt{1} = 0$$
  
  $$ -1 - 1 = -2 eq 0 $$ (Invalid)

By checking solutions, we confirm that only \( x = 3 \) is valid for this equation.