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Chapter 3: Problem 26

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$16 t^{2}-72 t+81=0$$

Short Answer

Expert verified
The solution to the equation is \( t = \frac{9}{4} \).

Step by step solution

01

Recognize the Quadratic Form

The given equation is in the standard form of a quadratic equation: \[ 16t^2 - 72t + 81 = 0 \] This can be identified by the general form of a quadratic equation \( ax^2 + bx + c = 0 \), where \( a = 16 \), \( b = -72 \), and \( c = 81 \).
02

Determine If It's a Perfect Square

Check if the quadratic can be expressed as a perfect square trinomial. A perfect square trinomial takes the form \( (pt + q)^2 \). We suspect this might be the case because of the symmetric coefficients.
03

Apply the Perfect Square Trinomial Formula

Let's assume the equation can be written as \[ (4t - 9)^2 = 0 \]Expand this to confirm:\[ (4t - 9)(4t - 9) = 16t^2 - 36t - 36t + 81 = 16t^2 - 72t + 81 \]This expansion matches the original equation.
04

Solve the Factored Equation

Since we established that \[ (4t - 9)^2 = 0 \], we can now solve for \( t \) by taking the square root of both sides:\[ 4t - 9 = 0 \]
05

Solve for t

Add 9 to both sides to isolate the term with \( t \):\[ 4t = 9 \]Then, divide both sides by 4:\[ t = \frac{9}{4} \]

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Key Concepts

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

Factoring Techniques
Factoring techniques are crucial in solving quadratic equations by simplifying them into a product of expressions. In the case of our quadratic equation, we aim to turn it into a product of two binomials. This simplifies finding the roots, as we can then solve for each factor set to zero. Understanding how to factor effectively requires recognizing when expressions represent patterns like difference of squares, perfect square trinomials, or simply common factors.
  • **Common Factors:** Look for numbers or variables that appear in each term. Factoring these out can simplify the equation.
  • **Difference of Squares:** Recognize expressions like 饾憥虏鈭掟潙徛� which can be factored into (饾憥鈭掟潙�)(饾憥+饾憦).
  • **Perfect Square Trinomials:** As seen in this example, quadratics like 16饾憽虏鈭�72饾憽+81 often resolve into squared binomials.
Developing a keen eye for these similarities can transform a quadratic equation into a simpler form, making it more straightforward to solve.
Perfect Square Trinomial
A perfect square trinomial is a special form of quadratic expression where the expression can be written as the square of a binomial. It's identified by the structure \[ ax^2 + 2abx + b^2 = (ax + b)^2 \] In this exercise, our equation 16饾憽虏鈭�72饾憽+81 meets this condition.
  • The term 16饾憽虏 equates to (4饾憽)虏, indicating the 'a' in the binomial (4饾憽鈭�9)虏.
  • The term 81 is the square of 9, indicating the 'b' in the binomial.
  • The middle term 鈭�72饾憽 is twice the product of 4饾憽 and 9, confirming the perfect square nature.
Recognizing these features allows for transforming the quadratic into \((4t - 9)^2 = 0\), simplifying the solving process down to a single linear factor. This takes advantage of symmetry and provides a powerful technique when applicable.
Solving Quadratic Equations
Solving quadratic equations involves finding the values of the variable that make the equation true. When equations are in their simplest factored form, such as \((4t - 9)^2 = 0\), solving becomes straightforward. ### Steps in Solving
  • **Identify Zero-Product Property:** If a product of factors equals zero, then at least one of the factors must be zero. This principle allows us to solve the factored equation \((4t - 9) = 0\).
  • **Solve the Linear Equation:** For our equation, set 4饾憽鈭�9 equal to zero and solve: add 9 to isolate 4饾憽 and divide by 4 to find 饾憽.
  • **Verify Your Solution:** It鈥檚 a good practice to substitute your solution back into the original equation to verify its correctness.
By leveraging the structure of the equation and simplifying through factoring, solving becomes a much clearer and more manageable process. This practice makes tackling quadratics both efficient and insightful.

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Most popular questions from this chapter

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$25 n^{2}+64$$

Set up an equation and solve each problem. The Ortegas have an apple orchard that contains 90 trees. The number of trees in each row is 3 more than twice the number of rows. Find the number of rows and the number of trees per row.

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Set up an equation and solve each problem. Suppose that the length of a certain rectangle is two centimeters more than three times its width. If the area of the rectangle is 56 square centimeters, find its length and width.

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$x^{2}+7 x+10=0$$
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