Problem 16 - Algebra A Combined Function

Question

Solve. $$ \sqrt{2 x-3}=3-x $$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Isolate the square root

To solve the equation \( \sqrt{2x - 3} = 3 - x \), we first ensure the square root is isolated on one side, which is already the case here.

Step 2: Eliminate the square root

Square both sides of the equation to get rid of the square root: \((\sqrt{2x - 3})^2 = (3 - x)^2\). Simplifying both sides gives us \(2x - 3 = 9 - 6x + x^2\).

Step 3: Rearrange into a quadratic equation

Bring all terms to one side to form a quadratic equation: \(x^2 - 8x + 12 = 0\).

Step 4: Solve the quadratic equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = -8\), and \(c = 12\). Calculate the discriminant: \((-8)^2 - 4 \cdot 1 \cdot 12 = 64 - 48 = 16\). Thus, \(x = \frac{8 \pm 4}{2}\).

Step 5: Find the potential solutions

Compute the potential solutions: \(x = \frac{8 + 4}{2} = 6\) and \(x = \frac{8 - 4}{2} = 2\). So, the potential solutions are \(x = 6\) and \(x = 2\).

Step 6: Check the solutions in the original equation

Substitute \(x = 6\) into the original equation: \(\sqrt{2(6) - 3} = 3 - 6 \Rightarrow \sqrt{9} = -3\), which is false. Thus, \(x = 6\) is not a solution. Substitute \(x = 2\) into the original equation: \(\sqrt{2(2) - 3} = 3 - 2 \Rightarrow \sqrt{1} = 1\), which is true. Thus, \(x = 2\) is a valid solution.

Key Concepts

quadratic formula

When solving certain types of equations, especially quadratic equations, the quadratic formula is a powerful tool. It helps to find the solutions of equations in the standard form of \[ ax^2 + bx + c = 0 \].The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here,

• \(a\) is the coefficient of \(x^2\),
• \(b\) is the coefficient of \(x\),
• and \(c\) is the constant term.

Simply plug your values of \(a\), \(b\), and \(c\) into the formula and solve. The plus or minus sign (\(\pm\)) indicates that there can be two solutions. This happens because of the square root part, which can produce both positive and negative values. Typically, these solutions are referred to as roots of the equation. Using this method is advantageous not only because it directly provides potential solutions, but it also confirms whether they are real by calculating the discriminant \(b^2 - 4ac\). A positive discriminant means two real and distinct solutions exist.

checking solutions in original equation

After finding potential solutions using the quadratic formula, it's crucial to check your solutions in the original equation. This step ensures that the solutions are valid for the specific equation you started with because sometimes the process of solving, such as squaring both sides, may introduce extraneous solutions.
Substitute each solution back into the original equation one by one. For example, if your potential solutions were \(x = 6\) and \(x = 2\), you would replace \(x\) in the original equation \[ \sqrt{2x - 3} = 3 - x \]with each value:

• For \(x = 6\), the equation \(\sqrt{2(6) - 3} = 3 - 6\) gives \(\sqrt{9} = -3\), which is not correct because the square root function cannot produce a negative number. So, \(x = 6\) is not a valid solution.
• For \(x = 2\), the equation \(\sqrt{2(2) - 3} = 3 - 2\) results in \(\sqrt{1} = 1\), which holds true. Thus, \(x = 2\) is the accurate and valid solution for the equation.

This checking procedure is essential to verify that due to simplifications made during solving, such as removing square roots, the original conditions are still satisfied.

eliminating square roots

When solving equations with square roots, the primary goal is to eliminate the square roots to simplify the equation. This often involves isolating the square root expression on one side of the equation and then squaring both sides. Consider the equation: \[ \sqrt{2x - 3} = 3 - x \]Here, the square root \(\sqrt{2x - 3}\) is already isolated, so the next step is to square both sides to remove the root:

• Squaring the left side: \( (\sqrt{2x - 3})^2 = 2x - 3 \)
• Squaring the right side: \( (3 - x)^2 = 9 - 6x + x^2 \)

This transformation gives us the quadratic equation \[ 2x - 3 = 9 - 6x + x^2 \],which can now be processed using algebraic manipulation to solve for \(x\). Remember that squaring both sides is a crucial step, but it may also introduce solutions that do not satisfy the original equation, hence it's important to later check the possible solutions obtained from the quadratic equation.