/*! 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 76 Solve the equation using any con... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 76

Solve the equation using any convenient method. $$4 x^{2}=7 x+3$$

Short Answer

Expert verified
The solutions to the quadratic equation are \(x = \frac{7 + \sqrt{97}}{8}\) and \(x = \frac{7 - \sqrt{97}}{8}\)

Step by step solution

01

Set the equation to standard form

The first step to solving a quadratic equation via the quadratic formula is to set the equation into standard form. The equation given is \(4x^{2}=7x+3\). Re-arranging the terms so that they equal zero, we get \(4x^{2} - 7x - 3 = 0\)
02

Identify a, b, and c

From the standard form, identify the values for a, b and c. From our equation \(4x^{2}- 7x - 3 = 0\), we can see that \(a = 4\), \(b = -7\), and \(c = -3\)
03

Substitute a, b, and c into the quadratic formula

We now substitute a, b, and c into the formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Substituting our values into the formula, we get, \(x = \frac{-(-7) \pm \sqrt{(-7)^{2} - 4*4*(-3)}}{2*4}\). Simplifying, we get, \(x = \frac{7 \pm \sqrt{49 - (-48)}}{8} = \frac{7 \pm \sqrt{97}}{8}\)
04

Simplify to find x

By simplifying the equation, we can find the values for x. We have two solutions for x, \(x = \frac{7 + \sqrt{97}}{8}\) and \(x = \frac{7 - \sqrt{97}}{8}\)

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Key Concepts

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

quadratic formula
The quadratic formula is a vital tool in algebra for finding the solutions to quadratic equations. These equations generally take the form of \(ax^2 + bx + c = 0\), where \(a eq 0\). The formula itself is expressed as:
  • \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
This formula helps to find the roots of any quadratic equation. The "\(\pm\)" symbol indicates that there will be two solutions, because we are considering both positive and negative square roots.
Some roots may be complex if the discriminant (\(b^2 - 4ac\)) is negative. The discriminant determines the nature of the roots, such as whether they are real and distinct, real and equal, or complex.
standard form
The standard form of a quadratic equation arranges all terms to one side of the equation, usually expressed as \(ax^2 + bx + c = 0\). In this form:
  • \(a\) represents the coefficient of \(x^2\)
  • \(b\) is the coefficient of \(x\)
  • \(c\) is the constant term
It is crucial to write the equation in this form to correctly identify the coefficients for use in the quadratic formula.

For instance, our equation \(4x^2 - 7x - 3 = 0\) is already in standard form, making it straightforward to define \(a = 4\), \(b = -7\), and \(c = -3\). Once in standard form, you can quickly transition to solving the problem using tools like the quadratic formula.
solving equations
Solving quadratic equations involves finding the values of \(x\) that make the equation true. These values are known as the roots of the equation. The process of solving:
  • Start by rearranging the equation into standard form if it is not already.
  • Identify \(a\), \(b\), and \(c\) based on the standard form.
  • Substitute those into the quadratic formula to solve for \(x\).
Simplifying the results gives the final values of \(x\).
For example, the equation \(4x^2 - 7x - 3 = 0\) gets solved by substituting \(a = 4\), \(b = -7\), and \(c = -3\) into the quadratic formula, leading to the solutions \(x = \frac{7 \pm \sqrt{97}}{8}\). Therefore, we have two possible values for \(x\), demonstrating how each part of the formula works together to produce the solution.
roots of equations
The roots of a quadratic equation are the solutions to the equation \(ax^2 + bx + c = 0\). These roots tell us at what values \(x\) will satisfy the equation.
In our context, the roots can be found using the quadratic formula. Once calculated, the roots can be represented as:
  • \(x = \frac{7 + \sqrt{97}}{8}\)
  • \(x = \frac{7 - \sqrt{97}}{8}\)
These roots may be real, repeated, or complex depending on the discriminant.
The calculation of the roots provides insight into the nature of the quadratic equation and where it crosses the x-axis on a graph. Understanding these helps to interpret broader patterns and behaviors in algebra.

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Most popular questions from this chapter

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$6\left(\frac{s}{s+1}\right)^{2}+5\left(\frac{s}{s+1}\right)-6=0$$

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the inequality \(W(t) \leq 600 .\) Explain what the solution of the inequality represents.

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|x-15|=x^{2}-15 x$$

You want to determine whether there is a relationship between an athlete's weight \(x\) (in pounds) and the athlete's maximum bench-press weight \(y\) (in pounds). Sample data from 12 athletes are shown below. (Spreadsheet at LarsonPrecalculus.com) (165,170),(184,185),(150,200), (210,255),(196,205),(240,295), (202,190),(170,175),(185,195), (190,185),(230,250),(160,150) (a) Use a graphing utility to plot the data. (b) A model for the data is \(y=1.3 x-36\).Use the graphing utility to graph the equation in the same viewing window used in part (a). (c) Use the graph to estimate the values of \(x\) that predict a maximum bench- press weight of at least 200 pounds. (d) Use the graph to write a statement about the accuracy of the model. If you think the graph indicates that an athlete's weight is not a good indicator of the athlete's maximum bench-press weight, list other factors that might influence an individual's maximum bench-press weight.

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=\sqrt{11 x-30}-x$$
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