Problem 10
Question
Solve the quadratic equation by factoring. $$x^{2}-16=0$$
Step-by-Step Solution
Verified Answer
The solutions to the equation \(x^{2}-16=0\) are \(x = 4\) and \(x = -4\)
1Step 1: Write the given equation
The provided quadratic equation is \(x^{2}-16=0\)
2Step 2: Rewrite the equation using the difference of squares identity
The given equation can be factored using the difference of squares identity \(a^{2}- b^{2} = (a - b)(a + b)\). Hence, the given equation \(x^{2}-16=0\) can be rewritten as \((x - 4)(x + 4) = 0\)
3Step 3: Set each factor equal to zero and solve for x
From Step 2 equation, to find the solutions, each factor is set to zero and then x is solved. So, \(x-4 = 0\) which gives \(x = 4\). And \(x+4 = 0\) which gives \(x = -4\)
Key Concepts
FactoringDifference of SquaresSolving EquationsAlgebraic Identities
Factoring
Factoring in mathematics refers to breaking down an expression into a product of simpler expressions. It is like finding pieces that multiply together to create the original expression. For a quadratic equation like \(x^2 - 16\), factoring can make solving for \(x\) much easier and more intuitive.
To factor a quadratic expression, you're often looking for pairs of numbers whose product gives you the constant term (in this case, \(-16\)) and whose addition or subtraction gives you the coefficient of the linear term (if it exists). In our example, since the linear term is missing, it simplifies to applying the **difference of squares identity** directly. This can immediately factor \(x^2 - 16\) into \((x-4)(x+4)\).
To factor a quadratic expression, you're often looking for pairs of numbers whose product gives you the constant term (in this case, \(-16\)) and whose addition or subtraction gives you the coefficient of the linear term (if it exists). In our example, since the linear term is missing, it simplifies to applying the **difference of squares identity** directly. This can immediately factor \(x^2 - 16\) into \((x-4)(x+4)\).
- Factoring reduces the complexity of solving equations.
- It often reveals solutions quickly by setting each factor to zero.
Difference of Squares
The difference of squares is a specific algebraic identity given by \(a^2 - b^2 = (a-b)(a+b)\). This identity is powerful because it allows you to rewrite expressions that are perfect squares with a subtraction sign into two binomials. Recognizing a difference of squares can significantly simplify complex algebraic expressions.
In the original exercise, \(x^2 - 16\) is structured as \(a^2 - b^2\), where \(a = x\) and \(b = 4\). Rewriting it using the difference of squares gives us \((x-4)(x+4)\).
This identity is especially useful in algebraic manipulations and solving quadratic equations efficiently.
In the original exercise, \(x^2 - 16\) is structured as \(a^2 - b^2\), where \(a = x\) and \(b = 4\). Rewriting it using the difference of squares gives us \((x-4)(x+4)\).
- Saves time, avoiding complex calculations.
- Makes it easier to identify potential equations to solve.
This identity is especially useful in algebraic manipulations and solving quadratic equations efficiently.
Solving Equations
Solving equations is about finding the value of the unknown variable that makes the equation true. In a quadratic equation like \(x^2 - 16 = 0\), the goal is to determine the values of \(x\) that satisfy this equation.
After factoring a quadratic equation, the next step is usually to set each factor equal to zero, which is based on the zero-product property: if a product of two numbers is zero, then at least one of the numbers must be zero. So, for \((x-4)(x+4) = 0\), we set \(x-4 = 0\) and \(x+4 = 0\).
Solving these simple equations will give the values of \(x\) as \(4\) and \(-4\).
After factoring a quadratic equation, the next step is usually to set each factor equal to zero, which is based on the zero-product property: if a product of two numbers is zero, then at least one of the numbers must be zero. So, for \((x-4)(x+4) = 0\), we set \(x-4 = 0\) and \(x+4 = 0\).
Solving these simple equations will give the values of \(x\) as \(4\) and \(-4\).
- Solving each factor leads directly to the solutions.
- Applying the zero-product property is a direct and effective method.
Algebraic Identities
Algebraic identities are formulas that represent the equality of two algebraic expressions. They are crucial tools for simplifying algebraic equations and can be used to factor expressions quickly.
In solving the equation \(x^2 - 16 = 0\), the difference of squares identity \(a^2 - b^2 = (a-b)(a+b)\) is an algebraic identity that allows precise factoring. Recognizing this pattern quickly simplifies the process of solving equational problems.
Applying such identities in algebra helps in maintaining accuracy and efficiency, enabling problem-solving in a simpler manner.
In solving the equation \(x^2 - 16 = 0\), the difference of squares identity \(a^2 - b^2 = (a-b)(a+b)\) is an algebraic identity that allows precise factoring. Recognizing this pattern quickly simplifies the process of solving equational problems.
- Bridges complex expressions to straightforward factorable forms.
- Reduces time required to factor and solve equations.
Applying such identities in algebra helps in maintaining accuracy and efficiency, enabling problem-solving in a simpler manner.
Other exercises in this chapter
Problem 10
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