Problem 21
Question
Solve each radical equation in Check all proposed solutions. $$ \sqrt{x+8}-\sqrt{x-4}=2 $$
Step-by-Step Solution
Verified Answer
The solution to the equation is \( x = 8 \)
1Step 1: Isolate one of the square roots
Get one of the square roots alone on one side of the equation by adding \(\sqrt{x-4}\) to both sides.So, we have: \( \sqrt{x+8} = \sqrt{x-4} + 2 \)
2Step 2: Square both sides
Square both sides to remove the square root:\[ (\sqrt{x+8})^2 = (\sqrt{x-4} + 2)^2 \]This simplifies to: \[ x + 8 = (x - 4 + 4\sqrt{x-4} + 4) \]Further simplification gives: \[ x + 8 = x + 4\sqrt{x-4} \]
3Step 3: Solve for x
Isolate x to one side of the equation, getting rid of any extra constants: \[ 0 = 4\sqrt{x-4} - 8 \]Divide each term by 4, finding:\[ 0 = \sqrt{x-4} - 2 \]Then, square both sides once more, giving:\[ 0 = x - 4 - 4 \]Which simplifies to: \[ x = 8 \]
4Step 4: Validate the solution
Substitute the calculated value of x into the original equation to verify the solution: \[ \sqrt{8+8}-\sqrt{8-4}=2 \]Simplifying, \( \sqrt{16}-\sqrt{4}=2 \)So, \( 4 - 2 = 2 \)which is a correct statement, hence the calculated value for x is a valid solution.
Key Concepts
Square RootsEquation SolvingVerification of Solutions
Square Roots
Square roots are a fundamental concept in mathematics. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 multiplied by 5 equals 25. In mathematical notation, the square root of a number is represented as \( \sqrt{} \).
Square roots can simplify numbers, helping to solve equations involving powers of two. They also enable us to work backwards from a number to find one of its two square roots.
In the context of solving radical equations, such as \( \sqrt{x+8} - \sqrt{x-4} = 2 \), knowing how to manipulate square roots is crucial. We can isolate one square root and solve by squaring both sides of the equation, helping us remove the roots altogether.
Square roots can simplify numbers, helping to solve equations involving powers of two. They also enable us to work backwards from a number to find one of its two square roots.
In the context of solving radical equations, such as \( \sqrt{x+8} - \sqrt{x-4} = 2 \), knowing how to manipulate square roots is crucial. We can isolate one square root and solve by squaring both sides of the equation, helping us remove the roots altogether.
Equation Solving
Equation solving is an essential skill in mathematics that involves finding the value of a variable that satisfies an equation. There are many techniques for solving equations, including isolation of terms and operations like addition, subtraction, multiplication, and division.
In solving radical equations, like our example, the strategy starts with isolating one of the square roots. This simplification step makes it easier to proceed by eliminating the square roots through squaring both sides of the equation. After squaring, the equation becomes a polynomial, where you can rearrange and solve for the variable by further simplifying.
It’s often necessary to square the equation more than once when additional square roots are present, as illustrated in the approach of removing another root to bring the problem further down to a basic algebraic level.
In solving radical equations, like our example, the strategy starts with isolating one of the square roots. This simplification step makes it easier to proceed by eliminating the square roots through squaring both sides of the equation. After squaring, the equation becomes a polynomial, where you can rearrange and solve for the variable by further simplifying.
It’s often necessary to square the equation more than once when additional square roots are present, as illustrated in the approach of removing another root to bring the problem further down to a basic algebraic level.
Verification of Solutions
Verification is a crucial step in solving equations! It ensures that the solutions obtained are actually correct. After solving the equation, you should substitute the solution back into the original equation to see if it holds true.
For instance, in our example, after finding \( x = 8 \), it's important to plug this value back into the initial equation \( \sqrt{x+8} - \sqrt{x-4} = 2 \). Doing so checks if both sides of the equation remain equal when \( x \) is replaced with 8.
This process may highlight extraneous solutions—values that emerged from algebraic manipulation steps but do not truly solve the original equation. Always validate to ensure that the obtained solutions are consistent throughout the initial problem requirements.
For instance, in our example, after finding \( x = 8 \), it's important to plug this value back into the initial equation \( \sqrt{x+8} - \sqrt{x-4} = 2 \). Doing so checks if both sides of the equation remain equal when \( x \) is replaced with 8.
This process may highlight extraneous solutions—values that emerged from algebraic manipulation steps but do not truly solve the original equation. Always validate to ensure that the obtained solutions are consistent throughout the initial problem requirements.
Other exercises in this chapter
Problem 21
Solve each quadratic inequality in Exercises \(1-28\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{2}
View solution Problem 21
Graph each equation in Exercises \(13-28 .\) Let \(x=-3,-2,-1\) \(0,1,2,\) and 3. $$y=|x|$$
View solution Problem 21
Solve each equation in Exercises \(15-26\) by the square root method. $$(3 x+2)^{2}=9$$
View solution Problem 21
In Exercises 13-26, express each interval in terms of an inequality and graph the interval on a number line. $$[-3, \infty)$$
View solution