Question

In Exercises 15–26, solve the equation. Check your solution(s). $$ \sqrt{2 x+30}=x+3 $$

Short answer

The solutions to the equation are x = 3 and x = -7, but only x = 3 is a valid solution when it's checked in the original equation.

Step-by-step solution

Isolate the Square Root

The best first step is to isolate the square root on one side. The equation is already isolated in that form: \( \sqrt{2 x+30}=x+3 \)

Square Both Sides

To get rid of the square root, the equation is squared on both sides of the equation. The squared equation should look like the following: \( (\sqrt{2x+30})^2=(x+3)^2 \)

Simplify the Equation

By doing the multiplication from the squaring, the equation is simplified: The square of \( \sqrt{2 x+30} \) is \( 2 x+30 \) on the left, and the square of \( x+3 \) is \( x^2+6x+9 \) on the right, so the new equation is: \( 2x+30=x^2+6x+9 \). Then, rearrange the equation to equal zero: \( x^2+4x-21=0 \).

Solving for x Using Quadratic Formula

To solve for x, the quadratic formula is used: \(x = (-b \pm sqrt{b^2-4ac}) / 2a\). Applying the formula gives two possible solutions for x.

Check the Solutions

To make sure these solutions are correct, substitute each one back into the original equation and make sure it holds true.

Key concepts

Isolation of Square Roots

Before solving an equation involving a square root, isolate the square root on one side. This means ensuring that all other terms in the equation are moved to the other side, leaving the square root by itself. In the given equation, \( \sqrt{2x+30} = x+3 \), the square root is already isolated.
The isolation step is crucial because it simplifies the process of dealing with the square root, making it easier to eliminate by squaring. Any other complex operations or variations in the equation are left for subsequent simplification once the square root is isolated.

Quadratic Formula

Sometimes, solving equations leads to a quadratic equation. These are equations in the form \( ax^2 + bx + c = 0 \). The quadratic formula is a reliable method to find solutions for these equations:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

For the quadratic equation derived from our problem, \(x^2 + 4x - 21 = 0\), identify:

• \(a = 1\)
• \(b = 4\)
• \(c = -21\)

Substitute these values into the quadratic formula to find the solutions for \(x\). This formula is essential since it guarantees real solutions if the discriminant \(b^2 - 4ac\) is non-negative.

Checking Solutions

After obtaining potential solutions, it's vital to check them in the original equation. This step ensures the solutions you’ve calculated are correct and applicable.
Substitute each solution back into the original equation \(\sqrt{2x+30} = x+3\) to determine if both sides of the equation match. If they don't, then that solution isn't valid.
This step prevents any extraneous solutions, which can arise, especially when dealing with squared equations. Always verify!

Simplifying Equations

Simplification is the process of breaking down expressions into their simplest form. When dealing with \( (\sqrt{2x+30})^2 = (x+3)^2 \), we square both sides to remove the square root, resulting in \(2x+30 = x^2 + 6x + 9\).
The next task is to rearrange this expression to form the quadratic equation: \(x^2 + 4x - 21 = 0\).
For simplification:

• Move all terms to one side to set the equation to zero.
• Combine like terms to minimize complexity.

This step ensures the equation is in a form that can be tackled using methods such as the quadratic formula. Simplifying equations is fundamental in algebra to make calculations manageable and solutions accessible.