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Chapter 8: Problem 67

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$900 x^{2}-8,100 x=1,800$$

Short Answer

Expert verified
The solutions are approximately \(x \approx 9.22\) and \(x \approx -0.22\).

Step by step solution

01

Write the Equation in Standard Form

Begin with the equation: \[ 900x^{2} - 8,100x = 1,800 \] Subtract \(1,800\) from both sides to set the equation to zero:\[ 900x^{2} - 8,100x - 1,800 = 0 \].
02

Simplify the Equation

Factor out the greatest common factor of the terms. In this case, the GCF is \(900\).Divide the entire equation by \(900\):\[ x^{2} - 9x - 2 = 0 \].
03

Identify and Use the Quadratic Formula

The equation \( x^{2} - 9x - 2 = 0 \) is in the standard form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -9 \), and \( c = -2 \). Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
04

Calculate the Discriminant

Calculate the discriminant using \(b^2 - 4ac\):\[ (-9)^2 - 4 \cdot 1 \cdot (-2) = 81 + 8 = 89 \].
05

Solve Using the Quadratic Formula

Substitute the values into the quadratic formula:\[ x = \frac{-(-9) \pm \sqrt{89}}{2} \]This simplifies to:\[ x = \frac{9 \pm \sqrt{89}}{2} \].
06

Approximate the Solutions

Calculate an approximate value for \( x \):Use a calculator to find \( \sqrt{89} \approx 9.43 \).Find the two solutions:\[ x = \frac{9 + 9.43}{2} \approx 9.215 \]and\[ x = \frac{9 - 9.43}{2} \approx -0.215 \].

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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 powerful tool for solving quadratic equations. It's particularly useful when factoring is challenging or cumbersome. A quadratic equation takes the form:
  • \( ax^2 + bx + c = 0 \)
Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). The quadratic formula provides the solutions for \( x \) as follows:
  • \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
The "+" and "-" signs in the formula indicate that there are typically two solutions. You can substitute the values of \( a \), \( b \), and \( c \) into this formula to find these solutions.
In our example equation:
  • \( a = 1 \)
  • \( b = -9 \)
  • \( c = -2 \)
Substituting these into the quadratic formula, we find the solutions step by step.
Discriminant
The discriminant is part of the quadratic formula under the square root sign:
  • \( b^2 - 4ac \)
It gives crucial information about the nature of the roots without having to solve the entire equation. In general:
  • If the discriminant is positive, there are two distinct real solutions.
  • If the discriminant is zero, there is exactly one real solution (repeated root).
  • If the discriminant is negative, there are no real solutions, but two complex solutions.
In our earlier example, the discriminant calculates to:
  • \( 81 + 8 = 89 \)
Since 89 is positive, this confirmed two distinct real solutions for the equation.
Standard Form
For any quadratic equation, writing it in standard form is an essential first step. The standard form is:
  • \( ax^2 + bx + c = 0 \)
This format makes it straightforward to identify the coefficients \( a \), \( b \), and \( c \), which are needed for both the quadratic formula and factoring. To transform a given equation into this form, rearrange all terms so that one side of the equation equals zero.
In our problem, the equation was initially:
  • \( 900x^2 - 8,100x = 1,800 \)
By moving \( 1,800 \) to the left side, we achieved the standard form:
  • \( 900x^2 - 8,100x - 1,800 = 0 \)
This is now suitable for further analysis or solving.
Factoring
Factoring is another method to solve quadratic equations, especially when the equation is simple or the roots are integers. The process involves expressing the quadratic equation as a product of two binomials. To factor, look for two numbers that multiply to give the constant term \( c \) and sum to give the linear coefficient \( b \).
In our exercise, the simplified equation is:
  • \( x^2 - 9x - 2 = 0 \)
However, this cannot be easily factored with integers, making the quadratic formula the preferred method here. It's important to note that factoring is efficient when the equations are suitable, but the quadratic formula remains universally applicable when factoring doesn't work.

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Most popular questions from this chapter

Explain why the quadratic formula, in most cases, is easier to use to solve a quadratic equation than is the method of completing the square.

The vertex of a quadratic function \(f(x)=a x^{2}+b x+c\) is given by the formula \(\left(-\frac{b}{2 a}, f\left(-\frac{b}{2 a}\right)\right) .\) Explain what is meant by the notation \(f\left(-\frac{b}{2 a}\right)\)

Find a number between 0 and 1 such that the difference of the number and its square is a maximum.

The number of female police officers (sworn) in the United States is approximated by the function \(f(t)=-24 t^{2}+1,534 t+72,065,\) where \(t\) is the number of years after \(2000 .\) In what year does the model indicate that the number of female officers reached \(75,000 ?\) .

Write each expression in radical form. $$ \left(3 m^{2} n^{2}\right)^{1 / 5} $$
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