Question

In Exercises 27–34, solve the equation. Check your solution(s). $$ (5-x)^{1 / 2}-2 x=0 $$

Short answer

The valid solution for the equation \((5-x)^{1 / 2}-2 x=0\) is \(x=1\)

Step-by-step solution

Isolate the square root

First arrange the equation and bring constant terms on one side, isolate the square root term on the other, so that the equation becomes \((5-x)^{1 / 2}=2x\)

Square both sides

Next square both sides of the equation to remove the square root: \( ((5-x)^{1 / 2})^2=(2x)^2 \), which simplifies to \(5-x=4x^2\)

Rearrange into standard quadratic form

Now rearrange the resulting equation into the standard form of a quadratic equation \(ax^2+bx+c=0\). This leads us to the equation \(4x^2+x-5=0\)

Solve the quadratic equation

Now, use the quadratic formula to solve the equation. The solutions are \(x1= 1\) and \(x2=-1.25\)

Check the solutions

Substitute the values of \(x1\) and \(x2\) into the original equation to verify if they are true solutions. The value \(x=1\) satisfies the original equation while the value \(x=-1.25\) doesn't. So the valid solution is \(x=1\)

Key concepts

Isolating the Square Root

Isolating the square root is the first crucial step in solving equations that include a square root function. The goal is to rearrange the equation so that the square root is by itself on one side of the equation.
To achieve this, start by moving other terms to the opposite side of the equation. This isolation helps make it easier to eliminate the square root in subsequent steps.
For example, in the equation \((5-x)^{1 / 2} = 2x\), we have isolated the square root term \((5-x)^{1 / 2}\) by moving other terms to the opposite side.
This manipulation sets the stage for the next step: squaring both sides to clear the square root.

Quadratic Formula

The quadratic formula is a reliable tool for solving quadratic equations and is especially handy when factoring is difficult or impossible. It is derived from the standard form of a quadratic equation:

• \( ax^2 + bx + c = 0 \)

The formula itself is given by:

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

In this context, the quadratic formula is used to find the values of \(x\) for the equation \(4x^2 + x - 5 = 0\).
By identifying \(a = 4\), \(b = 1\), and \(c = -5\), we plug these values into the quadratic formula, resulting in two potential solutions: \(x = 1\) and \(x = -1.25\).
This method provides a systematic way to solve any quadratic equation, ensuring that no potential solution is overlooked.

Checking Solutions

Checking solutions is an essential step in the process of solving equations, especially when dealing with square roots and quadratic equations. Once you have found the potential solutions using methods like the quadratic formula, you need to verify them by substituting back into the original equation.
For the potential solutions \(x = 1\) and \(x = -1.25\) found from the quadratic formula step, substituting these back into the original equation \((5-x)^{1 / 2} - 2x = 0\) helps us determine their validity.

• When \(x = 1\), the original equation holds true.
• However, when \(x = -1.25\), substituting it shows that the equation becomes invalid.

Thus, \(x = 1\) is confirmed as the only valid solution. This checking process ensures the accuracy of the solution and helps rule out any extraneous roots that might arise during the squaring step.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is an essential format used to simplify, analyze, and solve quadratic expressions. It is represented as:

• \( ax^2 + bx + c = 0 \)

Converting an equation into this standard form is a pivotal step before applying the quadratic formula.
For instance, after removing the square root by squaring both sides in our given problem, the equation \(5 - x = 4x^2\) is reorganized into the standard form \(4x^2 + x - 5 = 0\).
Each term in the equation has a specific role:

• \(ax^2\) is the quadratic term, defining the nature of the parabola.
• \(bx\) is the linear term, affecting the slope and direction.
• \(c\) is the constant term, shifting the curve up or down.

Organizing equations into standard form is not only crucial for solving them but also for understanding their properties and graphing them. Once in standard form, we can systematically solve the equation, ensuring we handle all variables appropriately and use the quadratic formula effectively.