Problem 64 - Algebra 2

Question

Solve \(\sqrt{x+2}=x\)

Step-by-Step Solution

Answer

By using the quadratic formula, we get \(x=2\) and \(x=-1\). After verification, only \(x=2\) satisfies the original equation and falls within the valid domain. Therefore, the solution to the problem \(\sqrt{x+2}=x\) is \(x=2\).

Step 1: Squaring both sides

To get rid of the square root, square both sides of the equation. This will yield \((\sqrt{x+2})^2=(x)^2\), which simplifies to \(x+2=x^2\)

Step 2: Bringing terms to one side

To make this equation solvable through the quadratic formula, we need it to be in the format of \(ax^2+bx+c=0\). So subtract \(x\) and 2 from both sides to get \(x^2-x-2=0\)

Step 3: Using the Quadratic Formula

Using the quadratic formula \(x = [-b±sqrt(b^2-4ac)] / 2a\), we solve \(x^2-x-2=0\) for \(x\). Here \(a=1, b=-1,c=-2\). Calculate \(x\) values using the quadratic formula.

Step 4: Verify solutions

The quadratic formula may yield valid solutions that do not actually satisfy the original equation after substitution, especially since we squared the equation at first, which sometimes introduces extraneous solutions. The equation \(\sqrt{x+2}=x\) is only defined for \(x≥-2\), so we need to verify the calculated values of \(x\) satisfy the original equation and fall within the valid domain.

Key Concepts

Solving Equations

When it comes to solving equations, it's essential to understand how to isolate variables and manipulate the equation to simplify it. In the context of quadratic equations like \(\sqrt{x+2}=x\), the goal is to handle the equation such that it leads us toward a straightforward solution. Initially, this equation is a radical equation because it involves a square root. To solve it, the technique of squaring both sides can be used, which helps eliminate the radical and rearrange the terms into a more familiar form. This allows us to work in the realm of polynomials, setting the stage for further solving techniques, like factoring or using formulas.

Square Roots

Square roots are an essential mathematical concept that often appears in equations. They can sometimes make an equation seem challenging due to their peculiar properties. In the exercise at hand, the equation initially had a square root on one side: \(\sqrt{x+2}=x\). To simplistically tackle this, square both sides. This converts the equation from a form that includes a radical into a polynomial equation, \(x+2 = x^2\).

• This step is crucial because it removes the complication the square root introduces.
• Once the equation becomes a polynomial, traditional methods like the quadratic formula can be employed.

Bear in mind that squaring both sides of an equation can sometimes introduce extraneous solutions, which is why validating potential solutions later is important.

Quadratic Formula

The quadratic formula is a powerful and universal method for solving quadratic equations of the form \[ax^2 + bx + c = 0\]. It allows us to find the values of \(x\) that make the equation true:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] In our problem, after simplifying to \(x^2-x-2=0\), we can apply the quadratic formula seamlessly.

• Here, \(a=1\), \(b=-1\), and \(c=-2\).
• Plug these values into the formula to find the roots of the equation.

After finding the roots, it's important to verify them in the original equation since squaring can introduce solutions that are not valid. Only solutions within the domain for which the original equation holds should be accepted.