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Chapter 1: Problem 84

Solve equation by the method of your choice. $$ 3 x^{2}-4 x=4 $$

Short Answer

Expert verified
The solutions of the equation \(3x^{2}-4x-4=0\) are \(x=2\) and \(x=-\frac{2}{3}\).

Step by step solution

01

Rewrite Equation as Standard Form

Start by re-writing the standard form \(ax^{2}+bx+c=0\), so \[3x^{2}-4x-4 = 0\] this will make applying the quadratic formula easier.
02

Apply Quadratic Formula

Once you have the equation in standard form with \(a=3\), \(b=-4\), and \(c=-4\), you can apply the quadratic formula \(\frac{-b卤\sqrt{b^{2}-4ac}}{2a}\), which will solve for \(x\). The formula will look like this: \[x=\frac{-(-4) \pm \sqrt{(-4)^{2}-4*3*(-4)}}{2*3}\]
03

Simplify Inside of the Square Root

Before proceeding with solving the equation, simplify the calculations inside the square root: \(16+48=64\). Then replace it back into the equation \[x=\frac{4卤\sqrt{64}}{6}\].
04

Solve for X

After you've simplified the square root, perform the final calculations, which gives you two potential answers for \(x\), using plus and minus operators: \[x=\frac{4+8}{6}=2\] and \[x=\frac{4-8}{6}=-\frac{2}{3}\]

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Key Concepts

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

Understanding Quadratic Equations
A quadratic equation is an essential concept in algebra. It's a type of polynomial equation distinguished by the highest power of its variable being 2. This gives the equation a characteristic curve shape known as a parabola when graphed. A basic quadratic equation is typically written as:
  • \(ax^2 + bx + c = 0\)
where:
  • \(a\): coefficient of \(x^2\), it cannot be zero.
  • \(b\): coefficient of \(x\).
  • \(c\): constant term.
Quadratic equations represent a wide range of phenomena, from projectile motion in physics to financial calculations in business. Understanding how to solve them is a fundamental skill.
Transforming to Standard Form
To effectively solve a quadratic equation, it's helpful to rewrite it in its standard form: \(ax^2 + bx + c = 0\). This rearrangement creates a clearer path for applying mathematical methods, such as completing the square or using the quadratic formula. Let's take an example:Given the equation: \(3x^2 - 4x = 4\),First, move all terms to one side of the equation to set it to zero:
  • Subtract 4 from both sides: \(3x^2 - 4x - 4 = 0\)
Now, it is in standard form, making it ready for applying various solution techniques. Working with equations in this format ensures consistency and correctness in further calculations.
Solving Equations Using the Quadratic Formula
The quadratic formula provides a straightforward method to find solutions to any quadratic equation. When the equation is in standard form \(ax^2 + bx + c = 0\), the quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Using this formula involves several important steps:
  • Calculate the values of \(b^2 - 4ac\), known as the discriminant.
  • The discriminant helps determine the nature of the roots.
  • Solve for \(x\) by simplifying the expression.
Let's solve the example equation \(3x^2 - 4x - 4 = 0\):
  • Here, \(a = 3\), \(b = -4\), and \(c = -4\).
  • First compute the discriminant: \((-4)^2 - 4 \cdot 3 \cdot (-4) = 64\).
  • Next, substitute these into the formula: \(x = \frac{4 \pm \sqrt{64}}{6}\).
  • This results in the solutions \(x = 2\) and \(x = -\frac{2}{3}\).
By following these steps, you can conclude that quadratic equations can be resolved effectively using the quadratic formula.

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Most popular questions from this chapter

In Exercises 142鈥�143, solve each inequality using a graphing utility. Graph each side separately. Then determine the values of x for which the graph for the left side lies above the graph for the right side. $$ -2(x+4)>6 x+16 $$

If a quadratic equation has imaginary solutions, how is this shown on the graph of \(y=a x^{2}+b x+c ?\)

A bank offers two checking account plans. Plan A has a base service charge of 4.00 dollar per month plus 10垄 per check. Plan B charges a base service charge of $2.00 per month plus 15垄 per check. a. Write models for the total monthly costs for each plan if x checks are written. b. Use a graphing utility to graph the models in the same [0, 50, 10] by [0, 10, 1] viewing rectangle. c. Use the graphs (and the intersection feature) to determine for what number of checks per month plan A will be better than plan B. d. Verify the result of part (c) algebraically by solving an inequality.

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. Parts for an automobile repair cost 175 dollar. The mechanic charges 34 dollar per hour. If you receive an estimate for at least 226 dollar and at most 294 dollar for fixing the car, what is the time interval that the mechanic will be working on the job?

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A city commission has proposed two tax bills. The first bill requires that a homeowner pay 1800 dollar plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of 200 dollar plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?
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