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Chapter 1: Problem 56

Find the real solutions, if any, of each equation. Use the quadratic formula. $$ 4 u^{2}-6 u+9=0 $$

Short Answer

Expert verified
The quadratic equation has no real solutions because the discriminant is negative.

Step by step solution

01

Identify Coefficients

Identify the coefficients in the quadratic equation \[ 4u^2 - 6u + 9 = 0 \] The coefficients are: \[ a = 4 \] \[ b = -6 \] \[ c = 9 \]
02

Quadratic Formula

Write down the quadratic formula \[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
03

Calculate the Discriminant

Calculate the discriminant \[ D = b^2 - 4ac \] Substitute the values for a, b, and c: \[ D = (-6)^2 - 4(4)(9) = 36 - 144 = -108 \]
04

Analyze the Discriminant

Since the discriminant \[ D = -108 \] is less than 0, the quadratic equation has no real solutions.

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Key Concepts

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

Real Solutions
In mathematics, real solutions are the values of the variable that satisfy a given equation and are real numbers (not imaginary or complex). When dealing with quadratic equations, we use the quadratic formula to find the solutions. A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \).

Generally, a quadratic equation can have:
  • Two real solutions
  • One real solution
  • No real solutions (if the solutions are complex)

To determine the nature of the solutions, we look at the discriminant (\( D \)).
This is essential in figuring out whether the solutions are real or complex.
In the example equation \( 4u^2 - 6u + 9 = 0 \), analyzing the discriminant reveals there are no real solutions.
Discriminant
The discriminant is a specific part of the quadratic formula that helps determine the nature of the solutions to the quadratic equation. It is found inside the square root in the quadratic formula: \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
The discriminant is represented by \( D \) and given by the formula: \( D = b^2 - 4ac \).

The value of the discriminant tells us the type of solutions:
  • If \( D > 0 \), there are two distinct real solutions.
  • If \( D = 0 \), there is exactly one real solution.
  • If \( D < 0 \), there are no real solutions; instead, there are two complex solutions.

In the equation \( 4u^2 - 6u + 9 = 0 \), the discriminant is calculated as \( D = (-6)^2 - 4(4)(9) = 36 - 144 = -108 \).
Since \( D < 0 \), this confirms that the equation has no real solutions.
Coefficients
Coefficients in a quadratic equation are the numerical factors that multiply the variables. They are essential in determining the properties and solutions of the equation. In the standard form of a quadratic equation \( ax^2 + bx + c = 0 \), the coefficients are:
  • \( a \): the coefficient of \( x^2 \)
  • \( b \): the coefficient of \( x \)
  • \( c \): the constant term or coefficient of \( x^0 \)

Identifying the coefficients is the first step in using the quadratic formula.
For the example \( 4u^2 - 6u + 9 = 0 \), we have:
  • \( a = 4 \)
  • \( b = -6 \)
  • \( c = 9 \)

These coefficients are crucial for determining the discriminant and ultimately solving the quadratic equation.
By substituting \( a, b, \) and \( c \) into the formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we can find the solutions step-by-step.

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Most popular questions from this chapter

Reducing the Size of a Candy Bar A jumbo chocolate bar with a rectangular shape measures 12 centimeters in length, 7 centimeters in width, and 3 centimeters in thickness. Due to escalating costs of cocoa, management decides to reduce the volume of the bar by \(10 \%\). To accomplish this reduction, management decides that the new bar should have the same 3 -centimeter thickness, but the length and width of each should be reduced an equal number of centimeters. What should be the dimensions of the new candy bar?

Find the real solutions, if any, of each equation. $$ 3 m^{2}+6 m=-1 $$

Find the real solutions of each equation. Use a calculator to express any solutions rounded to two decimal places. $$ x^{2 / 3}+4 x^{1 / 3}+2=0 $$

Challenge Problem Harmonic Mean For \(0

Find the real solutions, if any, of each equation. Use any method. $$ \frac{3 x}{x+2}+\frac{1}{x-1}=\frac{4-7 x}{x^{2}+x-2} $$
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