Problem 29 - College Algebra Essentials

Question

Check all proposed solutions. $$ \sqrt{3 \sqrt{x+1}}=\sqrt{3 x-5} $$

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{3 \sqrt{x+1}}=\sqrt{3x-5}\) is \(x = 4/3\)

Step 1: Simplify the both sides

The first step is to square the both sides of the equation to eliminate the square root. Doing this will give us \(3\sqrt{x+1}=3x-5\)

Step 2: Divide the both sides by 3

The next step would be to divide each side of the equation by 3 with the objective of further isolating x. This will give us \(\sqrt{x+1}=x-5/3\)

Step 3: Simplify the equation further

Take the equation from step 2 and square both sides to finally get rid of the remaining square root. Doing so gives you \(x+1=(x-5/3)^2\). This can be further simplified to \(x + 1 = x^2 - 10x/3 + 25/9\)

Step 4: Prepare for zero-product property

To use the zero-product property, one needs to ensure that the equation is equal to zero. Rearrange to get \(x^2 - 10x/3 + 25/9 - x - 1 = 0\). This simplifies to \(x^2 - 13x/3 + 16/9 = 0\)

Step 5: Apply Zero-product property

The Zero-product property states that if the product of two or more factors equals zero, then at least one of the factors must be zero. This is a general fact for all algebraic structures that includes a multiplication and an addition. Our equation from step 4 should now be factorable. \(x^2 - 13x/3 + 16/9 = 0\) can be factored to \((3x - 4)^2 = 0\)

Step 6: Solve for x

Lastly, solve for x to get the solution. That yields \(3x - 4 = 0\), and when you solve that for x, you end up with \(x = 4/3\) as your solution.

Key Concepts

Square Root

The concept of square root is crucial in algebra, particularly in solving quadratic equations. A square root such as \( \sqrt{a} \) refers to a number that, when multiplied by itself, gives \( a \) as the result. This is fundamental when dealing with equations involving square roots.
For example, if we have \( \sqrt{x} = y \), then \( y^2 = x \). In the given exercise, we started with the equation \( \sqrt{3 \sqrt{x+1}} = \sqrt{3x-5} \). To eliminate the square roots, both sides of the equation were squared, transforming it into a simpler form without square roots: \( 3\sqrt{x+1} = 3x-5 \).
This process highlights two steps in solving such equations:

• Square both sides to remove the square root.
• Maintain the equation's balance by performing the same operation on both sides.

Mastering these steps helps in effectively handling quadratic equations involving square roots.

Zero-product Property

The zero-product property is a powerful algebraic principle used to solve equations, particularly quadratics. This property states that if a product of two or more factors equals zero, at least one of the factors must be zero.
To apply this property effectively, the quadratic equation must be simplified and set to zero. For example, the equation \( x^2 - \frac{13x}{3} + \frac{16}{9} = 0 \) can utilize the zero-product property.
Factoring yields \((3x - 4)^2 = 0\). According to the zero-product property, since the product is zero, it means \(3x - 4 = 0\).
Applying this property involves:

• Rearranging the equation to equal zero.
• Factoring the equation.
• Setting each factor equal to zero to find the solutions.

These steps demonstrate how the zero-product property simplifies finding solutions in algebra.

Factoring Quadratics

Factoring quadratics is an essential skill in solving quadratic equations. This involves expressing a quadratic equation as a product of its linear factors. To apply this skill in solving an equation, the equation first needs to be simplified and set to zero.
This is seen in the step where we simplify \( x^2 - \frac{13x}{3} + \frac{16}{9} = 0 \). The equation can then be factored to \((3x - 4)^2 = 0\). The goal of factoring is to break down the quadratic into simpler terms that are easier to solve.

Here’s a breakdown of the factoring process:

• Rewrite the equation in standard form \( ax^2 + bx + c = 0 \).
• Find two numbers that multiply to \( a \times c \) and add to \( b \).
• Rewrite the middle term using these numbers, and then factor by grouping.
• Check the factored form by expanding it back to ensure correctness.

Factoring quadratics allows us to solve the equation efficiently, often leading directly to the solution.