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Chapter 9: Problem 85

Solve using the quadratic formula. $$ 9 x(x-1)+3(x+2)=1 $$

Short Answer

Expert verified
The quadratic equation has no real solutions.

Step by step solution

01

Expand the Equation

First, we expand the given equation to eliminate the parentheses:\[9x(x-1) + 3(x+2) = 1\]This becomes:\[9x^2 - 9x + 3x + 6 = 1\]
02

Simplify and Arrange into a Standard Quadratic Equation

Combine like terms and bring all terms to one side to get a standard quadratic equation:Combine like terms:\[9x^2 - 6x + 6 = 1\]Subtract 1 from both sides to set the equation to zero:\[9x^2 - 6x + 5 = 0\]
03

Identify Coefficients

From the quadratic equation \(9x^2 - 6x + 5 = 0\), identify the coefficients to use in the quadratic formula:\[a = 9, \quad b = -6, \quad c = 5\]
04

Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(x\):Substitute the coefficients into the formula:\[x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 9 \cdot 5}}{2 \cdot 9}\]Simplify under the square root:\[x = \frac{6 \pm \sqrt{36 - 180}}{18}\]
05

Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\):\[36 - 180 = -144\]Since the discriminant is negative, the 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.

Discriminant
In the realm of quadratic equations, the discriminant plays a crucial role in determining the types of solutions an equation may have. The discriminant is the part of the quadratic formula under the square root, represented as \(b^2 - 4ac\). Here are some key points about it:
  • If the discriminant is positive, the quadratic equation has two distinct real solutions.
  • If the discriminant is zero, there is exactly one real solution, known as a repeated or double root.
  • If the discriminant is negative, as in our example, the quadratic equation has no real solutions and instead has two complex solutions.
Understanding the discriminant helps quickly assess the nature of the solutions without fully solving the equation. In our exercise, the negative discriminant value of \(-144\) suggests that the solutions are not real numbers, thus no intersection points on the real number line.
Quadratic Equation
A quadratic equation is a polynomial equation of degree two, and it takes the standard form \(ax^2 + bx + c = 0\). In this form:
  • \(a\) is the coefficient of \(x^2\)
  • \(b\) is the coefficient of \(x\)
  • \(c\) is the constant term
Each quadratic equation graphs as a parabola on the coordinate plane, which may open upwards or downwards depending on the sign of \(a\). In our exercise, the equation was reformulated to \(9x^2 - 6x + 5 = 0\), with \(a = 9\), \(b = -6\), and \(c = 5\). By arranging in this form, we can easily identify the values necessary for calculation, especially when using the quadratic formula to find solutions.
Real Solutions
Real solutions to quadratic equations are the values of \(x\) that satisfy the equation and can be plotted on a real number line. These solutions are visible intercepts with the \(x\)-axis on the graph of a parabola. Here's how we determine them using the quadratic formula:The formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), incorporates the discriminant. Depending on the sign of the discriminant:
  • Positive discriminant indicates two real and distinct roots.
  • Zero discriminant signifies a single real root, where the parabola touches the \(x\)-axis just once.
  • Negative discriminant, which is our case here, outlines no real solutions, resulting in a parabola that doesn't cross the \(x\)-axis.
In the exercise, because the discriminant was \(-144\), we concluded the absence of real solutions. This means the solutions lie in the realm of complex numbers, beyond the scope of simple real intersections.

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