/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Solve each equation in Exercises... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 8

Solve each equation in Exercises \(1-14\) by factoring. $$ 4 x^{2}-13 x=-3 $$

Short Answer

Expert verified
The solutions to the equation \(4x^2 - 13x = -3\) are \(x = 3\) and \(x = \frac{1}{4}\)

Step by step solution

01

Rewrite the equation into standard quadratic form

To do this, add 3 to both sides of the given equation as follows: \(4x^2 - 13x + 3 = 0\)
02

Factorize the equation

To factorize the equation, search for two numbers that multiply to give 3 (the product of \(a\) and \(c\)) and add to give -13 (the value of \(b\)). The chosen numbers are -1 and -3. Therefore, \(4x^2 - 13x + 3 = 0\) could be factored as \(4x^2 - x - 12x + 3 = 0\) which is further grouped and factored as \((4x^2 - x) - (12x - 3) = 0\) to obtain \(x(4x - 1) - 3(4x - 1) = 0\). Hence, the quadratic equation could be represented as \((x - 3)(4x - 1) = 0\)
03

Solve the factored equation for \(x\)

Set each factor to zero and solve for \(x\) to get the solutions: \(x - 3 = 0\) yields \(x = 3\) and \(4x - 1 = 0\) yields \(x = \frac{1}{4}\). Therefore, the solutions of the equation are \(x = 3\) and \(x = \frac{1}{4}\)

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Key Concepts

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

Standard Quadratic Form
The standard quadratic form is essential for understanding how to solve quadratic equations. A quadratic equation is typically represented as ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants with 'a' not equal to zero. In our exercise example, we look at the equation 4x^2 - 13x = -3 and transform it into standard form by adding 3 to both sides, resulting in 4x^2 - 13x + 3 = 0.

This standard form is the starting point for solving quadratic equations and sets the stage for factoring. The constants 'a', 'b', and 'c' play a significant role in the process of factoring the equation, as they are used to find two numbers that will be pivotal in breaking down the equation into solvable pieces.
Solve Equations by Factoring
Factoring is a powerful method for solving quadratic equations. This approach involves expressing the quadratic equation as a product of two binomials. To solve an equation by factoring, we look for two numbers that multiply to the product of 'a' and 'c', and add to the value of 'b'.

In our exercise example, 4x^2 - 13x + 3 = 0 is factorized by finding two numbers that multiply to give 3 (4 * 3) and sum to -13. We successfully split the middle term, -13x, into two terms: -x and -12x, which are factored by grouping to yield the factored form (x - 3)(4x - 1) = 0. Factoring can be challenging, and it often helps to list out all the possible pairs of factors that could combine to reach the desired product and sum until the correct pairing is found.
Quadratic Equations Solutions
Once a quadratic equation has been factored, the next step is to find its solutions. This is done by setting each binomial factor equal to zero and solving for the variable x. If the product of two quantities equals zero, then at least one of the quantities must be zero.

Using the factored form from the previous concept, we set each factor equal to zero: (x - 3) = 0 and (4x - 1) = 0. The solutions are x = 3 and x = 1/4. These values are where the graph of the equation would intersect the x-axis. A quadratic equation can have a maximum of two real solutions, as represented by the highest power of x, which is 2. These solutions are also known as the 'roots' or 'zeros' of the equation.

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Most popular questions from this chapter

In a round-robin chess tournament, each player is paired with every other player once. The formula $$N=\frac{x^{2}-x}{2}$$ models the number of chess games, \(N,\) that must be played in a round-robin tournament with \(x\) chess players. Use this formula to solve Exercises \(131-132\). In a round-robin chess tournament, 36 games were played. How many players were entered in the tournament?

Will help you prepare for the material covered in the next section. $$\text{Rationalize the denominator: }\frac{7+4 \sqrt{2}}{2-5 \sqrt{2}}$$

In Exercises \(166-169\), determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I want to solve \(25 x^{2}-169=0\) fairly quickly, I'll use the quadratic formula.

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$S=\frac{C}{1-r} \text { for } r$$

The rectangular swimming pool in the figure shown measures 40 feet by 60 feet and is surrounded by a path of uniform width around the four edges. The perimeter of the rectangle formed by the pool and the surrounding path is 248 feet. Determine the width of the path.
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