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Chapter 9: Problem 22

Use the quadratic formula to solve each equation. $$ -6 a^{2}+3 a=-4 $$

Short Answer

Expert verified
The solutions are a =-( -2 o 3 ) and a =-( 2 o 3 ).

Step by step solution

01

Rewrite the equation in standard form

Rewrite the given equation oewlineo as , which is the standard quadratic form ewlineoewline Subtract 4 from both sides of the equation to get o.
02

Identify coefficients a, b, and c

From the standard form o, the coefficients are obtained directly: o (coefficient of ), o (coefficient of ), and o (constant term).
03

Apply the quadratic formula

The quadratic formula is o. Substitute the identified values of , , and into the formula to begin solving for .
04

Simplify under the square root

Calculate the discriminant o. Simplify the expression inside the square root carefully.
05

Compute the square root and simplify

Taking the square root of the simplified discriminant value and then completing the remaining operations in the quadratic formula provides the two solutions.

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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 an essential tool for solving quadratic equations. A quadratic equation has the general form \(a x^{2}+b x+c=0\). The quadratic formula allows us to find the values of \(x\) that satisfy the equation. The formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It is derived from the process called completing the square.
To use the quadratic formula, simply plug in the values of \(a\), \(b\), and \(c\) from your equation and then solve for \(x\). The plus-minus sign \( \pm \) means you will often get two solutions, one using the plus, and one using the minus.
standard form
A quadratic equation must be in standard form before using the quadratic formula. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). This form ensures that the equation is ready for the next steps of solving.
For example, given the equation \(-6 a^{2}+3 a=-4\), you need to rearrange it to \(-6a^{2} + 3a + 4 = 0\).
Now it matches \(ax^2 + bx + c = 0\). Remember, the standard form is crucial as it directly provides the coefficients \(a\), \(b\), and \(c\).
discriminant
The discriminant helps determine the nature of the solutions for a quadratic equation. It is part of the quadratic formula inside the square root: \(b^2 - 4ac\).
The value under the square root sign is called the discriminant and it plays a key role in finding the solutions.
The discriminant can tell us:
  • If it is positive, the quadratic equation has two distinct real roots.
  • If it is zero, there is exactly one real root (repeated).
  • If it is negative, the equation has two complex roots.
For the equation \(-6a^{2} + 3a + 4 = 0\), the discriminant is calculated as \(b^2 - 4ac = 3^2 - 4(-6)(4)\).
Simplify this carefully to find the discriminant value.
coefficients
The coefficients in a quadratic equation are the numerical constants multiplying the variables. In the standard form \(ax^2 + bx + c = 0\), the coefficients are designated as \(a\), \(b\), and \(c\).
They are essential for using the quadratic formula.
- \(a\) is the coefficient of \(x^2\).
- \(b\) is the coefficient of \(x\).
- \(c\) is the constant term.
Identifying these coefficients correctly is critical for solving equations.
For example, in \(-6a^{2} + 3a + 4 = 0\), we have \(a = -6\), \(b = 3\), and \(c = 4\).
Correctly using these in the quadratic formula ensures accurate solutions.

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Most popular questions from this chapter

Use the quadratic formula to solve each equation. $$ 2 m^{2}-12=6 m $$

Kai wants to create a fair dice game for two people, using divisibility by \(3 .\) He decided on these rules: \(\cdot\) Each player rolls one die. \(\cdot\) The players find the sum of the two numbers. \(\cdot\) Player 1 scores if the sum is divisible by \(3 .\) Player 2 scores if the sum is not divisible by \(3 .\) Kai isn't sure how many points each player should score each time. a. Make a table showing the probability of each sum being rolled. b. What is the probability that Player 1 scores on a given roll? c. Kai decided to give 2 points to Player 2 when the sum isn't divisible by \(3 .\) For the game to be fair, how many points should Player 1 score each time the sum is divisible by 3\(?\)

Five seventh grade friends鈥擜nya, Ben, Calvin, Dan, and Ezra鈥� challenged five eighth grade friends鈥擵ic, Wendi, Xavier, Yvonne, and Zac鈥攖o a backgammon tournament. They put the names of the seventh graders into one hat and the names of the eighth graders into another. To determine the two players for each match, they pulled one name from each hat. a. What is the size of the sample space in this situation? That is, how many different pairs of names are possible? Explain. b. What is the probability that the next match will involve Anya and either Xavier or Yvonne? c. What is the probability that the next pair drawn will not involve Calvin? d. Suppose all 10 names are put into one hat, and two are drawn at random. What is the probability that the pair will include one seventh grader and one eighth grader? Explain.

Factor each expression. $$ -4 k^{2}-5 k j-j^{2} $$

Sports A volleyball league has two divisions. At the end of the season, Team \(\mathrm{A}\) is in first place in Division 1 , followed by Team \(\mathrm{B}\) and Team \(\mathrm{C}\) . Team \(\mathrm{D}\) is in first place in Division 2 , followed by Team \(\mathrm{E}\) and Team \(\mathrm{F}\) . League organizers have structured the playoffs as shown below. a. Model this series six times, keeping a tally of the champion teams. Flip a coin or roll a die to determine the winner of each game. b. Which team won the most championships? c. Determine the probability that each team will win the champ-win. (Hint: Assume both teams in each game are equally likely to win. (Hint: Imagine that a series is played eight times, and determine how many times you can expect each team to win the championship.)
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