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Chapter 8: Problem 27

For each equation, determine what type of number the solutions are and how many solutions exist. $$y^{2}+\frac{9}{4}=4 y$$

Short Answer

Expert verified
Two distinct irrational solutions exist.

Step by step solution

01

- Move all terms to one side

First, move all terms to one side of the equation to set the equation to zero: \[ y^2 + \frac{9}{4} - 4y = 0 \]
02

- Simplify the quadratic equation

Rewrite the equation in standard quadratic form \(ay^2 + by + c = 0\). In this case: \[ y^2 - 4y + \frac{9}{4} = 0 \]
03

- Use the quadratic formula

To determine the solutions of the quadratic equation, use the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For the equation \( y^2 - 4y + \frac{9}{4} = 0 \), \(a = 1\), \(b = -4\), and \(c = \frac{9}{4}\).
04

- Calculate the discriminant

Calculate the discriminant \( \Delta = b^2 - 4ac \): \[ \Delta = (-4)^2 - 4 \cdot 1 \cdot \frac{9}{4} = 16 - 9 = 7 \]
05

- Determine the nature and number of solutions

Since the discriminant \( \Delta = 7 \) is positive and not a perfect square, there are two distinct irrational solutions.
06

- Solve for the solutions

Substitute \(a\), \(b\), and \(\Delta\) into the quadratic formula to find the solutions: \[ y = \frac{4 \pm \sqrt{7}}{2} \] Simplify to get: \[ y = 2 \pm \frac{\sqrt{7}}{2} \]

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Key Concepts

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

discriminant
The discriminant plays an important role in understanding quadratic equations. It is a value calculated from the coefficients of the quadratic equation and reveals the nature of the roots.
For a quadratic equation in the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is calculated as follows:
\[\Delta = b^2 - 4ac\]

The value of \(\Delta\) tells us different things about the solutions:
  • If \(\Delta > 0\), there are two distinct real solutions.
  • If \(\Delta = 0\), there is exactly one real solution (also called a repeated or double root).
  • If \(\Delta < 0\), the equation has two complex solutions.
The discriminant is essential for identifying the type and number of solutions without actually solving the equation.
quadratic formula
The quadratic formula is a tool for solving any quadratic equation \(ax^2 + bx + c = 0\). It provides the solutions directly from the coefficients of the equation without needing to factorize.
The quadratic formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Here's a step-by-step breakdown on how to use it:
  • Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation.
  • Calculate the discriminant \(\Delta = b^2 - 4ac\).
  • Substitute \(b\), \(a\), and \(\Delta\) into the quadratic formula.
  • Simplify the expression to find the solutions.
The quadratic formula is particularly useful when the quadratic equation does not factorize easily or when you need an accurate solution.
solution types
Quadratic equations can have different types of solutions based on the discriminant \(\Delta\). These solutions can be:
  • **Distinct Real Solutions**: If \(\Delta > 0\), there are two distinct real solutions. For instance, in the given equation, \(\Delta = 7\), which means it has two distinct irrational solutions: \(2 \pm \frac{\sqrt{7}}{2}\).
  • **Repeated Real Solution**: If \(\Delta = 0\), there is a single, repeated real solution. This happens when the parabola touches the x-axis at one point.
  • **Complex Solutions**: If \(\Delta < 0\), the equation has two complex solutions. In this case, the solutions involve imaginary numbers, appearing as conjugate pairs.
Understanding the type of solution is essential for interpreting the results of quadratic equations accurately. Each type of solution provides different insights into the behavior of the quadratic function.

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