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Chapter 8: Problem 8

For each equation, determine what type of number the solutions are and how many solutions exist. $$x^{2}-5 x+3=0$$

Short Answer

Expert verified
Two distinct real solutions

Step by step solution

01

Identify coefficients of the quadratic equation

For the quadratic equation in the form of \( ax^2 + bx + c = 0 \), identify the coefficients. Here, \(a = 1\), \(b = -5\), and \(c = 3\).
02

Find the discriminant

The discriminant \( \text{D} \) of a quadratic equation is given by the formula \( \text{D} = b^2 - 4ac \). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \( \text{D} = (-5)^2 - 4(1)(3) \).
03

Calculate the discriminant

Perform the calculations: \( \text{D} = 25 - 12 = 13 \).
04

Determine the type of roots based on the discriminant

Interpret the discriminant: If \( \text{D} > 0 \), the equation has two distinct real solutions. If \( \text{D} = 0 \), the equation has exactly one real solution. If \( \text{D} < 0 \), the equation has two complex solutions.
05

Conclusion

Since \( \text{D} = 13 \) is greater than 0, the quadratic equation has two distinct real solutions.

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Key Concepts

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

Discriminant
The discriminant is a crucial component in solving quadratic equations. It helps us understand the nature of the solutions without actually solving the equation. The discriminant is represented by the symbol \( \text{D} \) and is calculated using the formula:
\( \text{D} = b^2 - 4ac \).
Here, \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).
By calculating the discriminant, we can tell if the solutions will be real or complex, and how many solutions exist.
Real Solutions
Real solutions of a quadratic equation are the values of \( x \) that we obtain when we solve the equation. These solutions lie on the real number line. The number and nature of real solutions depend on the discriminant \( \text{D} \).
  • If \( \text{D} > 0 \), the equation has two distinct real solutions.
  • If \( \text{D} = 0 \), the equation has exactly one real solution.
  • If \( \text{D} < 0 \), the equation has no real solutions (the solutions will be complex).

In the given example, since the discriminant \( \text{D} = 13 \) is greater than 0, there are two distinct real solutions.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It allows us to find the solutions directly, using the coefficients from the equation. The formula is:
\[ x = \frac{-b \, \text{卤} \, \text{鈭殅 \text{(b}^2 \text{- 4ac})}{2a} \]
Here's how it works:
  • \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).
  • First, calculate the discriminant \( \text{D} \) using \( b^2 - 4ac \).
  • Then, plug \( \text{D} \), \( a \), and \( b \) into the quadratic formula.
  • The result will give the values of \( x \), which are the solutions to the equation.

Using this formula eliminates guesswork and provides a systematic approach to finding solutions.
Coefficients in Quadratic Equations
In any quadratic equation of the form \( ax^2 + bx + c = 0 \), the letters \( a \), \( b \), and \( c \) are called the coefficients. Each coefficient holds significant information:
  • \( a \) is the coefficient of \( x^2 \). It determines the parabola's direction (whether it opens upwards or downwards).
  • \( b \) is the coefficient of \( x \). It affects the location and shape of the parabola.
  • \( c \) is the constant term. It represents the y-intercept of the parabola.

In our example, the coefficients were:
  • \( a = 1 \) (opens upwards)
  • \( b = -5 \) (affects the slope and position)
  • \( c = 3 \) (y-intercept is 3)

Understanding these coefficients helps in manipulating and solving quadratic equations more efficiently.

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