Question

Solve the equation \(x^{2}-9=0\)

Short answer

The solutions are x = 3 and x = -3.

Step-by-step solution

Understand the Equation

The given equation is a quadratic equation in the form of x^{2} - 9 = 0. Notice that the equation is already in the form where one side is zero.

Factorize the Quadratic Expression

Recognize that x^{2} - 9 is a difference of squares. The difference of squares formula is a^2 - b^2 = (a - b)(a + b). Therefore: x^{2} - 9 = (x - 3)(x + 3).

Set Each Factor to Zero

To find the solutions, set each factor equal to zero and solve for x: (x - 3) = 0 or (x + 3) = 0.

Solve for x

Solve each simple equation: For (x - 3) = 0, add 3 to both sides: x = 3. For (x + 3) = 0, subtract 3 from both sides: x = -3.

Key concepts

Quadratic Equations

A quadratic equation is an equation that can be written in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero. In a quadratic equation, the highest power of the variable \(x\) is 2. Quadratic equations can take various forms and can be solved using several methods.
The exercise provided is already simplified to the form \(x^2 - 9 = 0\), making it easier to solve.
Understanding quadratic equations is crucial because they appear frequently in math problems related to physics, engineering, and finance. They can describe parabolic paths, areas, and more.
Always strive to simplify a quadratic equation to a form that sets one side to zero; this will make it easier to solve.

Difference of Squares

The difference of squares is a special type of quadratic expression. It takes the form: \(a^2 - b^2\). This type of expression can be factored conveniently using the formula: \(a^2 - b^2 = (a - b)(a + b)\).
The equation in the exercise, \(x^2 - 9\), is a perfect example of the difference of squares.
Here, \(a = x\) and \(b = 3\). So, \(x^2 - 9\) can be rewritten using the difference of squares formula as: \(x^2 - 3^2 = (x - 3)(x + 3)\).
This technique is a popular method for simplifying certain types of quadratic equations, making it easier to solve for the variable.
Recognizing when an equation fits this pattern will save you time and reduce complexity in solving quadratic equations.

Factoring Equations

Factoring is a method used to break down polynomials into simpler components called factors that, when multiplied together, give the original polynomial. To solve quadratics like \(x^2 - 9 = 0\), factoring is a powerful and efficient technique.
In our specific case, we already factorized \(x^2 - 9\) using the difference of squares formula.
The factored form is \((x - 3)(x + 3)\).
Factoring allows you to transform the quadratic equation from a polynomial expression into simpler binomials. This transformation simplifies the solving process significantly.
Always remember to check if the quadratic can be rewritten in a special form like the difference of squares or if other factoring methods (like finding common factors or using the quadratic formula) are more suitable.

Solving for x

Once you have factored a quadratic equation, the next step is to solve for the variable \(x\) by setting each factor equal to zero. This step is derived from the zero-product property, which states that if a product of multiple factors equals zero, at least one of the factors must be zero.
For \((x - 3)(x + 3) = 0\):

• Set the first factor equal to zero: \(x - 3 = 0\). Solve for \(x\): \(x = 3\).
• Set the second factor equal to zero: \(x + 3 = 0\). Solve for \(x\): \(x = -3\).

Thus, the solutions to \(x^2 - 9 = 0\) are \(x = 3\) and \(x = -3\).
This method can be applied to many quadratic equations, and understanding it will greatly benefit your problem-solving skills in algebra.