Problem 47 - Intermediate Algebra

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: Expand the Equation

Start by expanding the left side of the equation: \((x-3)^{2} = 12\)This can be expanded to:\(x^2 - 6x + 9 = 12\)

Step 2: Move all terms to one side

Subtract 12 from both sides to set the equation to zero:\(x^2 - 6x + 9 - 12 = 0\)This simplifies to:\(x^2 - 6x - 3 = 0\)

Step 3: Factor or Use Quadratic Formula

The quadratic equation \(x^2 - 6x - 3 = 0\) does not factor easily, so we apply the quadratic formula:\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=-6\), and \(c=-3\).

Step 4: Calculate the Discriminant

First, calculate the discriminant using the formula \(b^2 - 4ac\):\((-6)^2 - 4 \cdot 1 \cdot (-3) = 36 + 12 = 48\)

Step 5: Solve Using the Quadratic Formula

Plug the values into the quadratic formula:\(x = \frac{-(-6) \pm \sqrt{48}}{2 \cdot 1}\)Simplify to:\(x = \frac{6 \pm 4\sqrt{3}}{2}\)This results in:\(x = 3 \pm 2\sqrt{3}\)

Step 6: Write the Solutions

The solutions for the quadratic equation are:\(x = 3 + 2\sqrt{3}\) and \(x = 3 - 2\sqrt{3}\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. These types of equations take the form \( ax^2 + bx + c = 0 \). The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula helps find the roots or solutions of any quadratic equation. It's particularly useful when the quadratic cannot easily be factored.
To use the quadratic formula, identify the coefficients \(a\), \(b\), and \(c\) from the equation. Then plug these values into the formula.

• Step 1: Determine the values of \(a\), \(b\), and \(c\).
• Step 2: Plug the values into the formula.
• Step 3: Simplify the expression to find the two possible values for \(x\).

By following these steps, you can efficiently use the quadratic formula to solve any quadratic equation.

Discriminant

The discriminant is part of the quadratic formula. It's under the square root and is represented as \( b^2 - 4ac \). It gives important information about the nature of the roots of the quadratic equation.
Consider the following:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it's zero, there is exactly one real root, meaning the graph touches the x-axis at a single point.
• If it's negative, there are no real roots. Instead, you get two complex roots.

The discriminant helps you foresee the type of solutions to expect and decide which solving method might be best. For example, our example yielded a discriminant of 48, confirming two real and distinct solutions.

Factoring Quadratics

Factoring is a method to solve quadratic equations by expressing the quadratic expression as a product of two binomials. This method is often the quickest way to find solutions.
Not all quadratics factor easily, and this is when you might use other methods like the quadratic formula. However, when they do factor, this is a great tool:

• Step 1: Find two numbers that multiply to \(ac\) and add to \(b\).
• Step 2: Use these numbers to rewrite the middle term and factor by grouping.
• Step 3: Set each factor to zero and solve for \(x\).

In our original problem, \(x^2 - 6x - 3 = 0\), factoring was not straightforward. Therefore, the quadratic formula was utilized instead.

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial. This approach can be especially helpful in different contexts, such as graphing or deriving the quadratic formula.
To complete the square:

• Step 1: Make sure the coefficient of \(x^2\) is 1, which might involve dividing the equation.
• Step 2: Move the constant term to the opposite side of the equation.
• Step 3: Add the square of half the coefficient of \(x\) to both sides of the equation.
• Step 4: Write the left side as a square of a binomial.
• Step 5: Solve for \(x\) by finding the square root of both sides.

For instance, in the original problem \((x-3)^2 = 12\), it already looks like a square, which laid the ground for easy processing of the equation using square roots.