Problem 82 - Algebra for College Students

Question

Solve each equation. $$(5 x+6)^{\frac{1}{2}}=x$$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Square Both Sides

To eliminate the square root, square both sides of the equation. This gives: \[ (\sqrt{5x+6})^2 = x^2 \]which simplifies to:\[5x + 6 = x^2.\]

Step 2: Rearrange the Equation

Rearrange the equation to one side to form a standard quadratic equation:\[ x^2 - 5x - 6 = 0.\]

Step 3: Factor the Quadratic

To solve the quadratic equation, factor it. We look for two numbers that multiply to -6 and add to -5. The factors are -6 and 1, so the equation becomes:\[ (x - 6)(x + 1) = 0.\]

Step 4: Solve for x

Set each factor to zero to find solutions for \(x\):- \(x - 6 = 0\) gives \(x = 6\). - \(x + 1 = 0\) gives \(x = -1\).

Step 5: Check Each Solution in the Original Equation

Substitute \(x = 6\) back into the original equation:\[ \sqrt{5(6) + 6} = 6 \]\[ \sqrt{36} = 6 \]This is true, so \(x = 6\) is a valid solution.Substitute \(x = -1\) back into the original equation:\[ \sqrt{5(-1) + 6} = -1 \]\[ \sqrt{1} = -1 \]This is not true, so \(x = -1\) is not a valid solution.

Key Concepts

Solving Equations

Solving equations is a foundational skill in mathematics, focusing on finding the value(s) that satisfy the given equation. When tackling an equation, the first step is to understand what type of equation you're dealing with.
There are linear equations, quadratic equations, and more complex forms. In this exercise, our main goal is to solve a quadratic equation that appears after eliminating the square root.

• Identify the structure of the equation.
• Determine the operations needed to isolate the variable.

In our case, we started with an equation involving a square root: \((5x+6)^{\frac{1}{2}}=x\). By squaring both sides, we transformed it into a standard quadratic equation format, making it easier to solve. Always remember to check each solution to ensure it satisfies the original equation!

Factoring Quadratics

Factoring is a key method in solving quadratic equations and consists of expressing the equation as a product of two binomials set to zero. The standard form for a quadratic equation is \(ax^2 + bx + c = 0\). In the exercise above, after rearranging, we reached: \[x^2 - 5x - 6 = 0.\]
To factor this quadratic:

• Find two numbers whose product is the constant term, \(-6\), and whose sum is the coefficient of \(x\), which is \(-5\).
• The numbers that meet these conditions are \(-6\) and \(1\).

This allows us to rewrite the quadratic equation as:\((x - 6)(x + 1) = 0.\)
Factoring simplifies the solving process by breaking down a complex polynomial into simpler components that are easier to handle and solve.

Square Root Elimination

Eliminating a square root in an equation is an important technique, often necessary to simplify equations into more manageable forms.
When an equation involves a square root, such as \((5x+6)^{\frac{1}{2}}=x\), our objective is to remove the square root by applying squaring operations.

• Square both sides of the equation to eliminate the square root.
• For \(\left(\sqrt{5x+6}\right)^2\), the result is straightforward: \(5x + 6\).
• Ensure you apply the squaring operation to the entire expression on both sides to maintain equality.

Squaring both sides transforms the equation into a quadratic form \(x^2 = 5x + 6\), which then can be solved using factoring or other methods. It's crucial to check the solutions afterward, as squaring can introduce extraneous solutions, values which do not satisfy the original equation.