Question

Solve the equation. Check for extraneous solutions. $$x=\sqrt{11 x-28}$$

Short answer

The solution of the equation is \(x = 7\). The extraneous solution is \(x = 4\) and is discarded.

Step-by-step solution

Isolate the square root and square both sides

To eliminate the square root, let's square both sides of the equation. This gives us: \[(x)^2 =(\sqrt{11x - 28})^2 \] which simplifies to: \[x^2 = 11x - 28 \]

Solve the quadratic equation

Re-arrange this into the standard form of a quadratic equation \(ax^2 + bx + c = 0\) by subtracting \(11x\) and adding \(28\) to both sides. We have: \[x^2 -11x + 28 = 0 \] Now, we find factors of \(x^2 -11x + 28\) which add up to \(-11\) and multiply to \(28\). These factors are \(-7\) and \(-4\), so we can write the equation as: \[(x - 7)(x - 4) = 0\] Setting each factor equal to 0 gives the solutions \(x = 7\) and \(x = 4\].

Check for extraneous solutions

Let's substitute \(x = 7\) and \(x = 4\) into the original equation to verify if they are valid solutions or extraneous solutions. For \(x = 7\) we have: \(7 = \sqrt{11*7 - 28}\), which simplifies to \(7 = 7\). So, \(x = 7\) is a valid solution. For \(x = 4\) we have: \(4 = \sqrt{11*4 - 28}\), which simplifies to \(4 \neq 2\). So, \(x = 4\) is an extraneous solution, and we discard it.

Key concepts

Extraneous Solutions

When solving equations, particularly those involving square roots, we may encounter extraneous solutions. These are answers that emerge from the process of solving the equation but do not satisfy the original equation. They typically arise when we perform operations that are not reversible, such as squaring both sides of an equation to eliminate a square root. The process can introduce new solutions that were not solutions for the initial equation.

It is crucial to check all potential solutions by substituting them back into the original equation. If the substitution results in a true statement, the solution is valid. If not, the solution is extraneous and should be discarded. In our exercise, we spotted an extraneous solution when \(x = 4\) failed to satisfy the original equation after substitution.

Square Root Isolation

Isolation of the square root is a common strategy to tackle equations where the variable is under a square root sign. The goal is to get the square root expression by itself on one side of the equation. This simplifies matters and sets the stage for eliminating the square root by squaring both sides.

In the given exercise, we started by isolating the square root, \(\sqrt{11x - 28}\), on one side of the equation before squaring. It's important to perform this isolation carefully to ensure a smooth transition to the squaring stage. When isolating the square root, any misstep could complicate the equation further or produce errors, so attention to detail is paramount here.

Factoring Quadratics

Factoring quadratics is a fundamental skill in solving quadratic equations, where the goal is to rewrite the quadratic in a product of binomials. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). The factoring method involves finding two binomials that multiply to give the original quadratic and add to the middle term.

For the exercise, after rearranging the equation to the standard form, we factored \(x^2 - 11x + 28\) into \(x - 7\) and \(x - 4\). This step is vital because it reveals the potential solutions of the quadratic equation. When the product of two numbers equals zero, one or both of the numbers must be zero. We used this zero-product property to find the solutions. Naturally, factoring requires some practice to spot the correct binomial pairs quickly, a critical step towards solving these types of equations efficiently.