Problem 32 - Algebra and Trigonometry

Question

\(5-60\) Find all real solutions of the equation. $$ 2 x+\sqrt{x+1}=8 $$

Step-by-Step Solution

Answer

The real solution is \(x = 3\).

Step 1: Isolate the square root

Start by isolating the square root term on one side of the equation. To do that, subtract \(2x\) from both sides of the equation:\[\sqrt{x+1} = 8 - 2x\]

Step 2: Square both sides

Square both sides of the equation to eliminate the square root:\[(\sqrt{x+1})^2 = (8 - 2x)^2\]This simplifies to:\[x+1 = 64 - 32x + 4x^2\]

Step 3: Rearrange into a quadratic equation

Bring all terms to one side of the equation to form a quadratic equation:\[4x^2 - 33x + 63 = 0\]

Step 4: Use the quadratic formula

Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve the equation \(4x^2 - 33x + 63 = 0\), where \(a = 4\), \(b = -33\), and \(c = 63\):\[x = \frac{-(-33) \pm \sqrt{(-33)^2 - 4 \cdot 4 \cdot 63}}{2 \cdot 4}\]This simplifies to:\[x = \frac{33 \pm \sqrt{1089 - 1008}}{8}\]\[x = \frac{33 \pm \sqrt{81}}{8}\]\[x = \frac{33 \pm 9}{8}\]So, the possible solutions are \(x = \frac{42}{8} = 5.25\) and \(x = \frac{24}{8} = 3\).

Step 5: Verify the solutions

Check each potential solution by substituting back into the original equation:1. For \(x = 5.25\):\[2(5.25) + \sqrt{5.25 + 1} = 10.5 + \sqrt{6.25} = 10.5 + 2.5 = 13\]Since 13 does not equal 8, \(x = 5.25\) is not a solution.2. For \(x = 3\):\[2(3) + \sqrt{3 + 1} = 6 + \sqrt{4} = 6 + 2 = 8\]Since both sides of the equation are equal, \(x = 3\) is a valid solution.

Key Concepts

square root

The square root of a number is a value that, when multiplied by itself, gives the original number. In the equation we tackled, \(2x + \sqrt{x+1} = 8\), the square root term is \(\sqrt{x+1}\). When dealing with square roots in algebra, they can make equations a bit more complex. This is because their values are not always integers. By squaring both sides of an equation that contains a square root, we remove the square root and make it easier to solve.

To isolate the square root, you need to rearrange the equation so the square root is by itself on one side. Once isolated, you can square both sides:

• This gets rid of the square root
• It transforms the equation into a polynomial

This technique is crucial when solving equations with square roots, like the one we encountered.

isolating terms

Isolating terms in an equation means rearranging the equation so that one term is by itself on one side. This process helps simplify the equation and involves operations like addition, subtraction, multiplication, or division to move terms.

In our equation example, step 1 requires isolating the square root by subtracting \(2x\) from both sides, resulting in \(\sqrt{x+1} = 8 - 2x\). This is a classic algebraic technique to make an equation easier to handle. Doing this:

• Makes the variables clear
• Helps us focus on a single term or operation at a time

Isolating terms is the foundation to transforming and solving more complex equations, allowing the use of additional algebraic strategies.

quadratic formula

The quadratic formula is an essential tool in solving quadratic equations of the form \ ax^2 + bx + c = 0 \. It provides a straightforward method to find the roots of any quadratic equation. The formula is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\]
In our example, after simplifying the equation, it turns into a quadratic form: \(4x^2 - 33x + 63 = 0\). We applied the quadratic formula with \(a = 4\), \(b = -33\), and \(c = 63\).

This formula:

• Uses the coefficients directly from the quadratic equation
• Helps find solutions with both addition and subtraction of the square root

It's helpful wherever you encounter quadratic equations and ensures a systematic way to find possible solutions.

verifying solutions

Verifying solutions is the final and critical step in solving equations. Once we obtain potential solutions, it's necessary to substitute them back into the original equation. This checks if they truly satisfy the equation.

For the solved equation, when substituting \(x = 5.25\), we found that it did not satisfy the original equation \(2x + \sqrt{x+1} = 8\), so it was not a valid solution. However, substituting \(x = 3\) confirmed that both sides of the equation equaled 8. Thus, it was verified as a correct solution.

Verification ensures:

• The solutions are accurate
• Any extraneous solutions are identified

This step is crucial to ensure that the results reflect the real solutions to the problem.