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Chapter 7: Problem 29

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$0.29 Z^{2}-0.18=0.63 Z$$

Short Answer

Expert verified
The solutions are approximately \(Z_1 = 2.4281\) and \(Z_2 = -0.2557\).

Step by step solution

01

Rearrange the Equation

The original equation given is \(0.29 Z^2 - 0.18 = 0.63 Z\). First, rearrange the equation to match the standard form of a quadratic equation, \(aZ^2 + bZ + c = 0\). Move all terms to one side: \(0.29 Z^2 - 0.63 Z - 0.18 = 0\).
02

Identify Coefficients

Identify the coefficients \( a \), \( b \), and \( c \) from the rearranged equation \(0.29 Z^2 - 0.63 Z - 0.18 = 0\). So, \(a = 0.29\), \(b = -0.63\), and \(c = -0.18\).
03

Apply the Quadratic Formula

The quadratic formula is given by \(Z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 0.29\), \(b = -0.63\), and \(c = -0.18\) into the formula.
04

Compute the Discriminant

Calculate the discriminant part of the formula: \(b^2 - 4ac\). Substitute the values to get \((-0.63)^2 - 4 \times 0.29 \times (-0.18)\). First calculate \( (-0.63)^2 = 0.3969\), and \( 4 \times 0.29 \times (-0.18) = -0.2088\). Thus, the discriminant is \(0.3969 + 0.2088 = 0.6057\).
05

Evaluate the Square Root of the Discriminant

Compute the square root of the discriminant: \(\sqrt{0.6057}\). After calculation, \(\sqrt{0.6057} \approx 0.7783\).
06

Solve Using the Quadratic Formula

Substitute back into the quadratic formula: \(Z = \frac{-(-0.63) \pm 0.7783}{2 \times 0.29}\). Simplify to get \(Z = \frac{0.63 \pm 0.7783}{0.58}\).
07

Calculate the Two Possible Solutions

For the plus case, calculate \(Z_1 = \frac{0.63 + 0.7783}{0.58}\) which simplifies to approximately \( Z_1 = 2.4281\). For the minus case, calculate \(Z_2 = \frac{0.63 - 0.7783}{0.58}\) which simplifies to approximately \( Z_2 = -0.2557\).

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Key Concepts

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

Quadratic Formula
When faced with a quadratic equation of the form \(aZ^2 + bZ + c = 0\), the quadratic formula is your go-to tool for finding solutions. This magical formula provides the roots or solutions of any quadratic equation and is written as:
  • \(Z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Here's a quick breakdown:
  • \(b\) is the coefficient of the linear term, \(Z\).
  • \(a\) is the coefficient in front of the quadratic term, \(Z^2\).
  • \(c\) is the constant term.
This formula works in all scenarios--even if roots are not real numbers.
In cases where the discriminant is negative, you'll find imaginary roots.
Just like that, it gives you a reliable way to solve any quadratic equation, every time.
Discriminant
The discriminant is an essential component in understanding whether a quadratic equation has real solutions. It is the part of the quadratic formula located under the square root:
  • \(b^2 - 4ac\)

This value tells us different things based on its result:
  • If it is positive, there will be two distinct real roots.
  • If it equals zero, expect one real but repeated root.
  • If it is negative, then the equation has no real roots (instead, you will get complex solutions).
In our exercise:
  • Calculate \((-0.63)^2 - 4 \times 0.29 \times (-0.18) = 0.6057\).
  • Since 0.6057 is positive, there are two real roots, indicating a graph crossing the Z-axis at two points.
The discriminant gives you a sneak peek into the nature of the solutions before delving into deeper calculations.
Coefficients Identification
Successfully solving a quadratic equation requires pinpointing each term's coefficient in the expression \(aZ^2 + bZ + c = 0\). Let's take a closer look at each one:
  • \(a\) represents the coefficient of the \(Z^2\) term.
  • \(b\) is the coefficient attached to the \(Z\) term.
  • \(c\) is the constant or the term independent of \(Z\).

In the given exercise, properly identifying these coefficients is crucial:
  • With the equation \(0.29Z^2 - 0.63Z - 0.18 = 0\), we find that \(a = 0.29\), \(b = -0.63\), and \(c = -0.18\).
This identification allows us to directly substitute these values into the quadratic formula, ensuring accurate calculation of the roots or solutions. Always align the equation to its standard form as a first step, enabling easy coefficient recognition.

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Most popular questions from this chapter

Solve the given problems. All numbers are accurate to at least two significant digits. Find the smallest positive integer value of \(k\) if the equation \(x^{2}+3 x+k=0\) has roots with imaginary numbers.

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