Problem 47 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate. $$(x-3)^{2}=12$$

Step-by-Step Solution

Answer

The solutions are \(x = 3 + 2\sqrt{3}\) and \(x = 3 - 2\sqrt{3}\).

Step 1: Recognizing the Equation Type

The given equation \((x-3)^2 = 12\) is in the form of a perfect square trinomial equal to a number. This suggests using the square root method is appropriate.

Step 2: Apply the Square Root Method

To solve the equation, take the square root of both sides: \(\sqrt{(x-3)^2} = \sqrt{12}\). This simplifies to \(|x-3| = \sqrt{12}\).

Step 3: Simplify the Square Root

Simplify \(\sqrt{12}\) to \(2\sqrt{3}\), as 12 can be factored into \(4 \times 3\). Thus, \(|x-3| = 2\sqrt{3}\).

Step 4: Solve for x

This absolute value equation \(|x-3| = 2\sqrt{3}\) can be split into two separate equations: \(x-3 = 2\sqrt{3}\) and \(x-3 = -2\sqrt{3}\).

Step 5: Solve Each Equation

For \(x-3 = 2\sqrt{3}\):\[ x = 3 + 2\sqrt{3} \]For \(x-3 = -2\sqrt{3}\):\[ x = 3 - 2\sqrt{3} \] These provide the two solutions for \(x\).

Key Concepts

Square Root Method

The square root method is a handy technique for solving specific types of quadratic equations, especially when an equation is set to a perfect square trinomial on one side. When using the square root method, you'll take the following steps:

• Identify a perfect square trinomial in the equation.
• Take the square root of both sides of the equation.
• Keep in mind that the square root introduces an absolute value, indicating two potential solutions.

This method works best when the quadratic equation can be easily expressed in the form \((ax + b)^2 = c\). This simplifies the equation drastically, making it quick to solve.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be written as a square of a binomial. In simpler terms, it's something like \((x-a)^2\), which expands to \(x^2 - 2ax + a^2\). Spotting this pattern allows for straightforward simplification and use of the square root method.
To identify a perfect square trinomial:

• Check for the square of the first term \(x^2\).
• The middle term should be twice the product of the square root of the first and last terms.
• The last term should be the square of the constant.

In the problem \((x-3)^2=12\), \((x-3)^2\) is the perfect square trinomial.

Solving Quadratic Equations

Solving quadratic equations can often feel difficult, but understanding different methods makes it manageable. Quadratic equations can be solved by:

• Factoring, if the equation readily factors to two binomials.
• The quadratic formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).
• The completion of squares, converting the equation into the form \((x +- d)^2 = e\).
• The square root method, suitable for perfect square trinomials.

Choosing an appropriate method depends on the structure and complexity of the quadratic equation. Always start by understanding the equation's type to apply the best-suited method.

Absolute Value

Absolute value equations involve expressions where the solution could either be positive or negative, as you're dealing with the distance a number is from zero on a number line. When you encounter an equation with absolute value, such as \(|x-3| = 2\sqrt{3}\), it implies two separate cases:

• The expression inside the absolute value equals the positive counterpart: \(x-3 = 2\sqrt{3}\).
• Or the negative counterpart: \(x-3 = -2\sqrt{3}\).

Solving these individually gives you two possible solutions for \(x\). Remember, absolute value helps find both possible solutions that satisfy the original equation.