Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Solve the equation. $$ x+\sqrt{5 x+19}=-1 $$

Step-by-Step Solution

Answer

The solution is \(x = -3\), as it satisfies the original equation.

Step 1: Isolate the Square Root

We start by isolating the square root expression. To do so, subtract \(x\) from both sides of the equation: \(\sqrt{5x + 19} = -1 - x\).

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square root. This gives us: \(5x + 19 = (-1 - x)^2\).

Step 3: Expand and Simplify

Expand the right side of the equation: \((-1 - x)^2 = 1 + 2x + x^2\). So the equation becomes: \(5x + 19 = x^2 + 2x + 1\).

Step 4: Rearrange the Equation

Rearrange the equation to bring all terms to one side: \(x^2 - 3x - 18 = 0\).

Step 5: Solve the Quadratic Equation

We solve the quadratic equation \(x^2 - 3x - 18 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -3\), \(c = -18\).

Step 6: Calculate the Discriminant

The discriminant is \(b^2 - 4ac = (-3)^2 - 4(1)(-18) = 81\). Since the discriminant is positive, we have two real solutions.

Step 7: Obtain the Solutions

Plug the values into the quadratic formula: \(x = \frac{3 \pm \sqrt{81}}{2}\). Simplifying, we get \(x = \frac{3 + 9}{2} = 6\) and \(x = \frac{3 - 9}{2} = -3\).

Step 8: Verify the Solutions

Check both solutions in the original equation. For \(x = 6\), \(x + \sqrt{5x + 19} = 6 + \sqrt{49} = 6 + 7 = 13 eq -1\). Therefore, \(x = 6\) is not a solution. For \(x = -3\), \(x + \sqrt{5x + 19} = -3 + \sqrt{4} = -3 + 2 = -1\). This solution works.

Key Concepts

Square Root Isolation

When solving equations that contain square roots, the first crucial step is to isolate the square root component.
This means moving all other terms to the other side of the equation.
In our example equation, we start with

• \(x + \sqrt{5x + 19} = -1\)

To isolate the square root, subtract \(x\) from both sides:

• \(\sqrt{5x + 19} = -1 - x\)

Now the square root is by itself, making it easier to eliminate in the next steps.
Knowing how to properly manipulate an equation to isolate terms will be invaluable as you tackle more complex problems.

Discriminant Calculation

The discriminant is a powerful element of the quadratic formula that determines the nature of the solutions of a quadratic equation.
It's calculated from the formula:

• \(b^2 - 4ac\)

where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation standard form \(ax^2 + bx + c = 0\).
In our problem, the quadratic equation \(x^2 - 3x - 18 = 0\) has:

• \(a = 1\)
• \(b = -3\)
• \(c = -18\)

Substituting these into the discriminant formula provides:

• \((-3)^2 - 4(1)(-18) = 9 + 72 = 81\)

Because our discriminant is positive, we know there are two real solutions.Understanding the discriminant helps you predict the type of solutions without having to solve the entire equation.

Quadratic Formula

The quadratic formula is a staple tool for finding solutions to quadratic equations.
It is expressed as:

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

After calculating the discriminant, you substitute \(a\), \(b\), and \(c\) into the quadratic formula.
For our exercise, the substitutions are:

• \(a = 1\), \(b = -3\), \(c = -18\)

Therefore, we calculate:

• \(x = \frac{3 \pm \sqrt{81}}{2}\)
• \(x = \frac{3 \pm 9}{2}\)

These calculations yield the two possible solutions:

• \(x = 6\)
• \(x = -3\)

Using the quadratic formula is a reliable method to determine solutions, no matter how complex the quadratic.

Solution Verification

Verifying solutions is the final and crucial step in solving equations.
This step ensures that the values found actually satisfy the original equation.
For verification:

• Substitute each solution back into the original equation

For our equation \(x + \sqrt{5x + 19} = -1\):

• Checking \(x = 6\) gives: \(6 + \sqrt{49} = 6 + 7 = 13\), which does not equal -1. So, \(x = 6\) is not a solution.
• Checking \(x = -3\) gives: \(-3 + \sqrt{4} = -3 + 2 = -1\), which equals -1. Thus, \(x = -3\) is a valid solution.

Always verify your solutions as errors can occur during calculations, and only verified solutions ensure answer correctness.