Problem 35 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{-4 x+17}=x-3\)

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Eliminate the Square Root

Square both sides of the equation to eliminate the square root. \[ (-4x + 17) = (x - 3)^2 \]

Step 2: Expand and Simplify

Expand the right side and simplify the equation:\[ -4x + 17 = x^2 - 6x + 9 \]

Step 3: Rearrange Terms

Bring all terms to one side to set the equation to zero:\[ x^2 - 2x - 8 = 0 \]

Step 4: Factor the Quadratic Equation

Factor the quadratic equation:\[ (x - 4)(x + 2) = 0 \]

Step 5: Solve for x

Set each factor equal to zero:\[ x - 4 = 0 \] \[ x + 2 = 0 \] Solve to find \[ x = 4 \] and \[ x = -2 \] .

Step 6: Check Solutions

Check both potential solutions in the original equation to ensure they are valid.For \( x = 4 \):\[ \sqrt{-4(4) + 17} = 4 - 3 \]\[ \sqrt{1} = 1 \] (This is true, so \( x = 4 \) is a valid solution.)For \( x = -2 \):\[ \sqrt{-4(-2) + 17} = -2 - 3 \]\[ \sqrt{25} = -5 \] (This is false, as \(5 eq -5\), so \( x = -2 \) is not a valid solution.)

Key Concepts

Eliminating the Square Root

When solving equations involving a square root, the first step often involves eliminating the square root to simplify the equation. This process requires us to square both sides of the equation.
For example, if you have an equation like \( \sqrt{-4x + 17} = x - 3 \), squaring both sides removes the square root. So:

• On the left, \( \sqrt{-4x + 17} \) becomes \( -4x + 17 \).
• On the right, \( (x - 3)^2 \) remains \( x^2 - 6x + 9 \).

By squaring to eliminate the square root, we convert the problem into one involving polynomial expressions, making it simpler to solve. However, be careful as squaring can introduce extra solutions that may not satisfy the original equation.

Factoring Quadratic Equations

Factoring is a classic technique used to solve quadratic equations. After expanding and simplifying the equation, we often deal with quadratic forms such as \( ax^2 + bx + c = 0 \).
The goal is to express this form as a product of two binomials. For example, with the equation \( x^2 - 2x - 8 = 0 \):

• Look for two numbers that multiply to \(-8\) and add up to \(-2\): these are \(-4\) and \(2\).
• Factor the equation as \((x - 4)(x + 2) = 0\).

Using the Zero Product Property, we set each factor equal to zero to find potential solutions.

Expand and Simplify

The step of expanding and simplifying involves transforming parts of the equation into a more workable format. Here, it's important to handle polynomial expressions carefully by applying the distributive property, also known as FOIL for binomials.
For instance, when expanding \((x - 3)^2\):

• Apply the distributive property: \( x^2 - 3x \) and \(-3x + 9 \).
• Combine like terms: simplify to \( x^2 - 6x + 9 \).

After this, the equation \(-4x + 17 = x^2 - 6x + 9\) further involves moving terms to combine them into a standard quadratic form \( x^2 - 2x - 8 = 0\). Rearrangement to form a single polynomial equation is crucial for further solving steps.

Checking Solutions

Once potential solutions are found, it's vital to substitute them back into the original equation to verify their validity. Solutions that do not satisfy the original equation are extraneous and should be discarded.
For the example given:

• Check each solution by plugging it back: for \(x = 4\), substituting back proves valid as both sides equal 1.
• For \(x = -2\), the check results in a false statement \(5 eq -5\), hence this solution is invalid.

This checking step ensures that only valid solutions which satisfy the original equation are considered, highlighting the importance of verification in solving equations.