Problem 22 For Exercises 5 to $62,$ solve... [FREE SOLUTION]

Chapter 10: Problem 22

For Exercises 5 to $62,$ solve. $$16 x^{2}-1=0$$

Short Answer

Expert verified

Therefore, the solutions to the equation $16x^2 - 1 = 0$ are $x = 1/4$ and $x = -1/4$

Step by step solution

Recognize the formula

The given equation is $16x^{2} -1=0$. Recognize it as the form of $a^{2}-b^{2}$. Where $a = 4x$ and $b = 1$ because $a^{2} = (4x)^{2} = 16x^{2} $ and $b^{2} =(1)^{2} =1$

Factorize the equation using the difference of squares formula

We now rewrite the equation in the factorised form using the difference of squares formula. That gives us $(a - b)(a + b) = 0$ which now becomes $( 4x - 1)( 4x + 1) = 0 $

Solve for x

Now, split up the equation and equate each part to 0. So, either $4x-1 = 0$ or $4x+1 = 0$. On solving the these two equations, we get $x = 1/4$ and $x = -1/4$ respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares

The concept of the difference of squares is fundamental in solving quadratic equations. It refers to a specific type of algebraic expression where you subtract one squared term from another. For example, in our exercise, the equation is presented as:

$16x^2 - 1 = 0$

This can be rearranged to reveal its structure as a difference of squares:

$(4x)^2 - 1^2 = 0$

Here, $4x$ and $1$ are terms that, when squared, give back $16x^2$ and $1$ respectively. The difference of squares formula is given by:

$a^2 - b^2 = (a - b)(a + b)$

In this form, we factor the expression into two binomials. It's essential to identify when a quadratic expression fits this pattern because it allows you to simplify and solve equations more efficiently.

Factoring

Factoring is a technique used to rewrite algebraic expressions as the product of simpler expressions. When we apply factoring to a difference of squares, it involves expressing it in the form of two binomials. Let's see how this works with the equation:

Start with $16x^2 - 1$.
Recognize it as $(4x)^2 - (1)^2$.
Factor using the formula $a^2 - b^2 = (a - b)(a + b)$.

This means we rewrite the expression as follows:

$(4x - 1)(4x + 1) = 0$

Factoring transforms the quadratic equation, making it easier to find the solutions. By breaking down the equation into two linear expressions, you open up straightforward paths for solving the values of $x$. Factoring is not only used for difference of squares, but it鈥檚 a versatile tool for multiple kinds of polynomials.

Solving Equations

Once an equation has been factored, solving it becomes a matter of finding the values of the variable that make each factor equal to zero. From the example, after factoring, our equation becomes:

$(4x - 1)(4x + 1) = 0$

To solve for $x$, we apply the zero product property, which says if a product of factors equals zero, at least one of the factors must be zero:

$4x - 1 = 0$
$4x + 1 = 0$

Solving each, we find:

For $4x - 1 = 0$:
- Add 1 to both sides: $4x = 1$.
- Divide by 4: $x = \frac{1}{4}$.
For $4x + 1 = 0$:
- Subtract 1 from both sides: $4x = -1$.
- Divide by 4: $x = -\frac{1}{4}$.

This approach is not limited to difference of squares but is a general method in quadratic equations. By factoring and solving each part separately, you can uncover all possible solutions swiftly and accurately.

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91影视

For Exercises 5 to \(62,\) solve. $$16 x^{2}-1=0$$

Short Answer

Step by step solution

Recognize the formula

Factorize the equation using the difference of squares formula

Solve for x

Key Concepts

Difference of Squares

Factoring

Solving Equations

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