Problem 18 - Algebra 2

Question

Solve each equation. $$ \sqrt{2 t-7}=\sqrt{t+2} $$

Step-by-Step Solution

Answer

The solution is \( t = 9 \).

Step 1: Eliminate the Square Roots

To solve the equation \( \sqrt{2t - 7} = \sqrt{t + 2} \), start by eliminating the square roots on both sides by squaring each side of the equation. This gives us:\[(\sqrt{2t - 7})^2 = (\sqrt{t + 2})^2\]which simplifies to:\[ 2t - 7 = t + 2 \]

Step 2: Rearrange the Equation

Next, rearrange the equation \( 2t - 7 = t + 2 \) to get all terms involving \( t \) on one side and the constant terms on the other. Subtract \( t \) from both sides:\[ 2t - t - 7 = t - t + 2 \]which simplifies to:\[ t - 7 = 2 \]

Step 3: Solve for t

Now solve for \( t \) by adding 7 to both sides of the equation \( t - 7 = 2 \):\[t = 2 + 7\]which simplifies to:\[t = 9\]

Step 4: Verify the Solution

To ensure that \( t = 9 \) is correct, substitute it back into the original equation \( \sqrt{2t - 7} = \sqrt{t + 2} \):- For the left-hand side: \( \sqrt{2(9) - 7} = \sqrt{18 - 7} = \sqrt{11} \).- For the right-hand side: \( \sqrt{9 + 2} = \sqrt{11} \).Since both sides evaluate to \( \sqrt{11} \), the solution is verified.

Key Concepts

Square Roots

Square roots are a crucial concept in mathematics. They allow us to find a number that, when multiplied by itself, results in the given number. For example, the square root of 9 is 3 because 3 multiplied by 3 gives 9. In equation solving, square roots can often be part of the expression, as seen in the example \( \sqrt{2t - 7} = \sqrt{t + 2} \). To effectively solve such equations, it is necessary to remove the square root using a technique called "squaring both sides."

• This means you will multiply each side of the equation by itself.
• This step transforms the equation into a form without square roots, making it simpler to solve.

Squaring both sides should be done with caution. It's essential to remember that squaring can introduce extraneous solutions—solutions that are mathematically correct but do not actually satisfy the original equation. This is why verification after solving is a necessary step.

Verification of Solutions

Verification of solutions is the process of checking if the solution really satisfies the original equation. It involves plugging the obtained solution back into the original equation to see if both sides of the equation are equal. Using the earlier example, if we find \( t = 9 \) as the solution, we substitute back: - Substitute \( t = 9 \) into \( \sqrt{2t - 7} \) to get \( \sqrt{2(9) - 7} = \sqrt{11} \)- Similarly, substitute \( t = 9 \) into \( \sqrt{t + 2} \) to get \( \sqrt{9 + 2} = \sqrt{11} \)

• If both calculations result in the same value, the solution is verified.
• If not, there might be an extraneous solution due to the process of squaring both sides.

This extra step ensures accuracy and confirms that your solution is correct and applicable to the original problem.

Quadratic Equations

Quadratic equations are a fundamental part of algebra. They are polynomial equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. These equations can have various methods of solutions, such as factoring, using the quadratic formula, or completing the square. In the context of our square root equation, once the square roots were removed, we were left with a simpler linear equation: \[ 2t - 7 = t + 2 \]Although this is not a quadratic equation itself, understanding the transformation from a radical equation helps in comprehending quadratic forms when you encounter them.

• Quadratic equations can often be created from such transformations, especially when dealing with more complicated expressions that involve squaring both sides.
• When solving quadratic equations, always check for both possible solutions since they can result from different factors in the equation.

Just as with simple linear equations, precise steps and verification ensure correct interpretation and application of solutions.