Problem 45 - Algebra 2

Question

Solve each equation by using the method of your choice. Find exact solutions. \(21=(x-2)^{2}+5\)

Step-by-Step Solution

Answer

The solutions are \(x = 6\) and \(x = -2\).

Step 1: Isolate the quadratic term

First, subtract 5 from both sides of the equation to isolate the quadratic term on one side:\[ 21 - 5 = (x - 2)^{2} \]This simplifies to:\[ 16 = (x - 2)^{2} \]

Step 2: Take the square root of both sides

To solve for \(x\), we take the square root of both sides of the equation. Remember to include both the positive and negative roots:\[ \pm \sqrt{16} = x - 2 \]This becomes:\[ x - 2 = 4 \] or \[ x - 2 = -4 \]

Step 3: Solve for x

Now solve for \(x\) in each case. For \(x - 2 = 4\):\[ x = 4 + 2 \]\[ x = 6 \]For \(x - 2 = -4\):\[ x = -4 + 2 \]\[ x = -2 \]

Step 4: Conclusion: State the solutions

The solutions to the equation are the values of \(x\) we found in the previous step: \( x = 6 \) and \( x = -2 \).

Key Concepts

Quadratic Formula

When faced with quadratic equations, the quadratic formula is a universal tool that allows us to find solutions, known as "roots". It's applicable to any quadratic equation that has the structure: \[ ax^2 + bx + c = 0 \].
The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula helps us find the values of \(x\) that make the quadratic equation true.
Here's how it works:

• Identify the coefficients: \(a\), \(b\), and \(c\) from your equation.
• Calculate the discriminant, which is \(b^2 - 4ac\). This part of the formula determines the nature of the roots. When it's positive, you have two distinct real solutions.
• Use the formula by substituting the known values. The \(\pm\) sign indicates you'll get two solutions: one by adding the square root of the discriminant and one by subtracting it.

The quadratic formula is a powerful method especially when factoring isn't straightforward or possible.

Square Roots

Square roots are an essential concept when working with quadratic equations. When you take the square root of a number, you are finding a value that, when multiplied by itself, will give you the original number.
In equations like \( (x - 2)^2 = 16 \), the square root is used to "undo" the squaring.
Here's a step-by-step practice:

• Isolate the squared term, like \( (x-2)^2 \).
• Take the square root of both sides of the equation to eliminate the square. Remember to account for both the positive and negative outcomes. For example, \( \pm \sqrt{16} \) means you consider both +4 and -4 as possible results.

Square roots are simple in this exercise, but it's crucial to remember their dual nature when solving equations.

Exact Solutions

Exact solutions are important in mathematics because they provide precise answers without any approximation. In solving a quadratic equation, finding the exact solution involves getting values of \(x\) that precisely satisfy the quadratic equation.
In the steps provided, we used the square root method to discover the two exact solutions: \(x = 6\) and \(x = -2\).
Here's why these solutions are exact:

• They were derived directly from the equation using algebraic manipulation, maintaining all integers and without rounding.
• Exact solutions typically hold all decimal places as per the mathematical operations involved in solving the equation. This makes them very reliable particularly when precision is necessary, such as in scientific calculations.

By verifying, you can substitute these values back into the original equation and both should satisfy the equation perfectly.