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Chapter 6: Problem 101

Solve each equation. $$ d(8 d-9)=-1 $$

Short Answer

Expert verified
The solutions are \(d = 1\) and \(d = \frac{1}{8}\).

Step by step solution

01

Expand the Equation

First, we'll expand the equation by distributing the term outside the parentheses:\[ d(8d - 9) = -1 \]This becomes:\[ 8d^2 - 9d = -1 \]
02

Set the Equation to Standard Form

Next, we will rearrange the equation to set it into the standard form of a quadratic equation:\[ 8d^2 - 9d + 1 = 0 \]
03

Use the Quadratic Formula

The quadratic formula is given by:\[ d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]For this equation, \( a = 8 \), \( b = -9 \), and \( c = 1 \). Plug in these values:\[ d = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \times 8 \times 1}}{2 \times 8} \]
04

Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \):\[ (-9)^2 - 4 \times 8 \times 1 = 81 - 32 = 49 \]
05

Solve for d using the Quadratic Formula

Substitute the discriminant back into the quadratic formula:\[ d = \frac{9 \pm \sqrt{49}}{16} \]\[ d = \frac{9 \pm 7}{16} \]Thus, solving gives two solutions:\( d = \frac{16}{16} = 1 \) and \( d = \frac{2}{16} = \frac{1}{8} \).

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Key Concepts

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

Quadratic Formula
A quadratic equation is characterized by its highest exponent being 2 and usually appears in the form \( ax^2 + bx + c = 0 \). The quadratic formula is a powerful tool used to find the solutions of such equations. It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's what each part means:
  • \( a \) refers to the coefficient of the \( x^2 \) term.
  • \( b \) refers to the coefficient of the \( x \) term.
  • \( c \) is the constant term.
To use the quadratic formula, follow these steps:
  • Rewrite the equation in the standard form: \( ax^2 + bx + c = 0 \).
  • Identify the coefficients \( a \), \( b \), and \( c \).
  • Substitute these coefficients into the quadratic formula.
  • Solve for \( x \) using the values obtained from the formula.
By substituting appropriately, you can find the roots or solutions of any quadratic equation.
Discriminant
The discriminant is a vital concept in understanding the nature of solutions for a quadratic equation. It is the part under the square root in the quadratic formula, given by:\[\Delta = b^2 - 4ac\]The discriminant helps determine the number and type of solutions:
  • If \( \Delta > 0 \), there are two distinct real solutions.
  • If \( \Delta = 0 \), there is exactly one real solution, a repeated root.
  • If \( \Delta < 0 \), there are no real solutions; instead, there are two complex solutions.
Every value of the discriminant offers insight into the quadratic's graph, such as whether it touches or crosses the x-axis.When solving a quadratic equation as shown in the previous exercise, calculating the discriminant tells us what kind of roots to expect before further solution. In our case, with \( \Delta = 49 \), we have two distinct real solutions.
Standard Form of a Quadratic Equation
Getting a quadratic equation into its standard form is the first step to utilizing tools like the quadratic formula. The standard form is:\[ ax^2 + bx + c = 0 \]The equation needs:
  • \( ax^2 \): the term with the squared variable.
  • \( bx \): the linear term.
  • \( c \): the constant term.
Transforming an equation into this form involves moving all terms to one side of the equation, so that 0 remains on the other side.For example, in the exercise, we began with:\( 8d^2 - 9d = -1 \)By adding 1 to both sides, we nested it into:\( 8d^2 - 9d + 1 = 0 \)Once in standard form, identifying the coefficients \( a \), \( b \), and \( c \) becomes straightforward, allowing us to proceed to deeper analysis and solutions.

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Most popular questions from this chapter

Factor using rational numbers. $$ x^{6}-4 x^{3}-12 $$

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ x^{3}-25 $$

Graph the interval \((-2,3]\) on a number line.

What does it mean to factor a polynomial?

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 5(x-2)+10 y(x-2) $$
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