Problem 19 - Algebra and Trigonometry

Question

Solve each radical equation in Exercises 11–30. Check all proposed solutions. $$\sqrt{2 x+19}-8=x$$

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{2 x+19}-8=x\) is \(x = -5\).

Step 1: Isolate the radical

Add 8 to both sides of the equation to move 8 from the right hand side, so we get \(\sqrt{2 x+19}=x+8\)

Step 2: Square both sides to remove the square root

This gives \((\sqrt{2x+19})^2=(x+8)^2\). When we simplify it we get \(2x+19 = x^2 + 16x +64\).

Step 3: Rearrange the equation

Get rid of x on the left and move the constant to the other side. Therefore the equation look like this: \(x^2 +16x -2x +64 -19 =0\). Then simplify and we get \(x^2 +14x +45 =0\)

Step 4: Solve for x

Solve this quadratic equation by factorising. Therefore the factors of 45 that add up to 14 are 9 and 5. So this becomes \((x+9)(x+5)=0\). This give us \(x_1 =-9\) and \(x_2 = -5\) as potential solutions to the equation.

Step 5: Substitute the solutions

Let's check each potential solution by plugging it into the original equation. After substituting the values it's clear that -9 is not a solution because the result inside the square root would be negative. Plugging in -5 we get that the left-hand side equals the right-hand side, hence \(-5\) is a solution to the equation.

Key Concepts

Isolate the Radical

When solving radical equations, the first important step is to isolate the radical expression. This means getting the term with the square root by itself on one side of the equation. For instance, in the given equation \(\sqrt{2x+19} - 8 = x\), the radical is \(\sqrt{2x+19}\). To isolate it, we need to move the -8 to the other side.

This is done by adding 8 to both sides of the equation, resulting in:

• \(\sqrt{2x+19} = x + 8\)

After this step, the radical expression \(\sqrt{2x+19}\) stands alone on the left side, ready for the next steps in solving the equation.

Quadratic Equation

Once the radical is isolated, the next step involves removing the square root. We do this by squaring both sides of the equation, aiming to eliminate the square root and deal with a simpler form. With the isolated radical \(\sqrt{2x+19} = x + 8\), squaring both sides gives us:

• \((\sqrt{2x+19})^2 = (x + 8)^2\)
• This simplifies to: \(2x + 19 = x^2 + 16x + 64\)

After squaring, we're left with a quadratic equation: \(x^2 + 16x - 2x + 64 - 19 = 0\). Simplifying it further results in \(x^2 + 14x + 45 = 0\).

Now, this is a standard quadratic equation form, which can be solved using methods like factoring, completing the square, or applying the quadratic formula. Here, we choose to factor for simplicity.

Factorization

Factorization is a method used to solve quadratic equations by expressing the quadratic as a product of two binomials. In the quadratic equation \(x^2 + 14x + 45 = 0\), our goal is to find two numbers that multiply to 45 (the constant term) and add up to 14 (the coefficient of the linear term).

The numbers 9 and 5 satisfy these conditions, so we write the quadratic as:

• \((x+9)(x+5)=0\)

By setting each binomial equal to zero, we solve for the potential solutions of \(x\):

• \(x+9=0\) gives \(x=-9\)
• \(x+5=0\) gives \(x=-5\)

These are the potential solutions of the quadratic equation, but we must check each one to ensure they actually solve the original radical equation.

Check Solutions

After solving a radical equation, it's crucial to check the solutions because squaring both sides can introduce extraneous solutions—solutions that don't actually satisfy the original equation. We substitute each found solution back into the initial equation to verify its validity.

For our solutions, \(x = -9\) and \(x = -5\), plug them back into the original radical equation \(\sqrt{2x+19} - 8 = x\):

• For \(x = -9\): Plugging it in gives a negative value under the square root, which is not possible. So, \(-9\) is not a valid solution.
• For \(x = -5\): Substitution shows that both sides of the equation equal, confirming \(-5\) as a valid solution.

Checking solutions is a crucial final step in solving radical equations to ensure all proposed solutions are correct.