Problem 14 - College Algebra Essentials

Question

Check all proposed solutions. $$ \sqrt{x+10}=x-2 $$

Step-by-Step Solution

Answer

The solution for the equation \(\sqrt{x+10}=x-2\) is \(x=6\).

Step 1: Isolate the square root term

Keep the square root term \(\sqrt{x+10}\) on the left side and move the rest to the right giving \(\sqrt{x+10}=x-2\).

Step 2: Square both sides

Next, raise both sides of the equation by the power of 2 (i.e, square both sides) to remove the square root, thereby getting: \((\sqrt{x+10})^2=(x-2)^2\), which reduces to \(x+10 = x^2 - 4x + 4\).

Step 3: Simplify to obtain a quadratic equation

Subtract \(x\) and \(10\) from both sides to transform the equation into a standard quadratic equation, which gives us: \(x^2 - 5x - 6 = 0\).

Step 4: Factor the quadratic equation

Factor the quadratic equation into two binomials. This gives: \((x - 6)(x + 1) = 0\).

Step 5: Solve for \(x\)

Set each factor equal to zero and solve for \(x\). This gives potential solutions \(x=6\) or \(x=-1\).

Step 6: Check your solutions

Substitute these values back into the original equation to verify their validity. Upon substitution, it is found that \(x=6\) is a valid solution whereas \(x=-1\) is an extraneous solution - it does not satisfy the original equation. Therefore, discard \(x=-1\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra, helping you find the unknown value that makes an equation true. To solve an equation, you perform operations that simplify the equation, with the goal of isolating the unknown variable. For the equation \[ \, \sqrt{x+10} = x-2\,\,, \]we aim to find the value of \(x\) that satisfies this equality.Here’s a handy approach:

• Start by isolating terms with the unknown variable, just like moving terms around to make the equation more manageable.
• Then perform operations like adding, subtracting, multiplying, or dividing both sides of the equation by the same value. Doing the same thing to both sides keeps the equation balanced.

In this example, the square root is isolated on one side. By solving step by step, we ensure accuracy and ease.

Extraneous Solutions

Extraneous solutions are solutions that appear valid but don't satisfy the original equation.As we solve equations, especially ones involving square roots or fractions, sometimes we end up with solutions that don't actually work when plugged back into the starting equation. It's crucial to check all potential solutions in the original equation to confirm their validity.In our problem:

• We found potential solutions \(x = 6\) and \(x = -1\) by solving the quadratic equation.
• However, when replacing \(x = -1\) back into the original equation, it doesn't hold true. This makes \(x = -1\) an extraneous solution.

Always verify your answers by substituting them back into the original equation to confirm they work, discarding the ones that don't.

Factoring Quadratics

Factoring quadratics is a method used to solve second-degree polynomial equations that take the form \(ax^2 + bx + c = 0\).This technique involves expressing the quadratic equation as a product of two binomials. For the quadratic equation \(x^2 - 5x - 6 = 0\), we factor it to:\[(x - 6)(x + 1) = 0\]Factoring involves finding two numbers whose product equals \(-6\) (the constant \(c\)) and whose sum equals \(-5\) (the coefficient \(b\)).

• In this instance, the numbers \(-6\) and \(1\) work perfectly.
• Once factored, set each binomial equal to zero to find the solutions.

Factoring is a powerful tool because it turns complex polynomials into simpler expressions, making it easier to find solutions for quadratic equations.

Square Root Equations

Square root equations involve an unknown variable under a square root sign. Solving them often requires isolating the square root expression, then squaring both sides to eliminate the root.Consider the equation:\[ \,\sqrt{x+10}=x-2\,,\,, \]
To solve, isolate the root and then square both sides:

• When you square both sides: \[(\sqrt{x+10})^2 = (x-2)^2 \]
• This simplifies, removing the square root: \[ x+10 = x^2 - 4x + 4 \]

Squaring is a common step in solving square root equations, but it's essential to check for extraneous solutions, as squaring can introduce values that don't work in the original equation.