Question

Write the quadratic equation in general form. $$ (x-3)^{2}=2 $$

Short answer

The quadratic equation in general form is \(x^2 - 6x + 7 = 0\).

Step-by-step solution

Expand the Equation

Expand the left side of the equation \( (x-3)^{2} = 2 \). This means calculating \((x - 3)^2 = x^2 - 6x + 9 \). So the equation now is \(x^2 - 6x + 9 = 2\).

Simplify the Equation

Now, simplify the equation by subtracting 2 from both sides to maintain the balance. The simplified equation becomes \(x^2 - 6x + 9 - 2 = 0\), which further simplifies to \(x^2 - 6x + 7 = 0 \). This is the quadratic equation in standard form.

Key concepts

Expanding Quadratic Equations

When dealing with quadratic equations, one of the fundamental skills is to expand them from their factored or presented form into the standard quadratic form. This often involves applying the FOIL method (First, Outer, Inner, Last) or using the square of a binomial formula. For instance, take the equation (x-3)^2=2. To expand this, you would multiply (x-3) by itself, which results in x^2 - 6x + 9. Essentially, you're transforming the equation into a form that reveals the a, b, and c constants of the general quadratic equation ax^2 + bx + c = 0.

Why is this important? Expanding equations makes it possible to identify these constants and proceed with solving the equation by methods such as factoring, completing the square, or using the quadratic formula. Remember, a thorough expansion avoids common algebraic mistakes, so take your time and check each step for accuracy.

Simplifying Algebraic Expressions

After expanding a quadratic equation, you'll often be left with an expression that can be further simplified. Simplification makes the equation easier to solve and understand. The process usually involves combining like terms and performing basic arithmetic operations to condense the expression. With our previous example, we initially expanded (x-3)^2 to get x^2 - 6x + 9 = 2. The next step, simplification, is about maintaining the equation's balance as we aim to achieve the standard form.

In this case, subtracting 2 from both sides gives us x^2 - 6x + 7 = 0, with no further like terms to combine. The equation is now primed for solving. Students must take care not to alter the equation's values inadvertently by ensuring that every action performed on one side is mirrored on the other. This is crucial for preserving the equation's integrity.

Quadratic Equations Steps

Solving quadratic equations typically follows a series of methodical steps. Starting with the expanded and simplified form, the aim is to find the values of x that satisfy the equation.

The steps in solving a quadratic equation often include:

• Expanding any factored parts of the equation, as we did in our example, to reveal the full quadratic expression.
• Simplifying the equation by combining like terms and bringing all terms to one side of the equation to set it equal to zero.
• Applying a method of solving quadratics, such as factoring, using the quadratic formula, or completing the square. The choice of method is typically guided by the nature of the quadratic's coefficients and constants.

Each of these steps is crucial and skipping one can lead to incorrect solutions. Understanding the logic behind each step is key to mastering quadratic equations, not just following the process mechanically. Remember, practice makes perfect, so work through several examples to become adept at recognizing the most efficient method for each scenario.