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Chapter 10: Problem 76

Tell how many solutions the equation has. $$ 3 x^{2}-7 x+5=0 $$

Short Answer

Expert verified
The quadratic equation \(3x^{2}-7x+5=0\) has no real solutions

Step by step solution

01

Identify the coefficients a, b, and c

The quadratic equation is in the form \(ax^{2}+bx+c=0\). Here, \(a=3\), \(b=-7\), and \(c=5\).
02

Calculate the discriminant

Use the formula \(b^{2}-4ac\) to calculate the discriminant. So here, it is \((-7)^{2}-4*3*5 = 49-60 = -11\)
03

Determine the number of solutions

As the discriminant is less than 0 (-11), it implies that the quadratic 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 world of quadratic equations, the discriminant plays a crucial role in determining the nature of the solutions. The discriminant is derived from the formula \(b^{2} - 4ac\), where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^{2} + bx + c = 0\).
The value of the discriminant indicates whether the solutions are real numbers or complex numbers.
Here's how the discriminant tells us about the solutions:
  • If the discriminant is positive, the equation has two real and distinct solutions.
  • If the discriminant is zero, the equation has exactly one real solution, also known as a repeated or double root.
  • If the discriminant is negative, there are no real solutions; instead, the solutions are complex or imaginary.
In our example \((-11)\), since the discriminant is negative, it confirms the absence of real solutions.
Real Solutions
Real solutions in quadratic equations indicate that the solutions are real numbers, which are numbers that can be located on the number line. They include all the rational and irrational numbers.
Quadratic equations can have zero, one, or two real solutions:
  • Two distinct real solutions occur when the graph of the quadratic equation crosses the x-axis at two points, a scenario possible when the discriminant is positive.
  • One real solution occurs when the graph of the equation is tangent to the x-axis, meaning it just touches the axis at one point. This happens when the discriminant is zero.
  • No real solutions occur if the graph does not intersect the x-axis at all, corresponding to a negative discriminant. This results in complex solutions instead.
In the given exercise, the quadratic equation has no real solutions because the graph doesn't cross the x-axis due to a negative discriminant.
Quadratic Formula
The quadratic formula is an invaluable tool for solving quadratic equations. It's given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula can find solutions to any quadratic equation, whether or not the equation has real solutions.
Here's how it functions:
  • First, calculate the discriminant \(b^2 - 4ac\). The solutions depend on the value of this discriminant.
  • Take the square root of the discriminant. If the discriminant is positive, this will yield real numbers; if zero, there鈥檚 only one solution; if negative, it involves complex numbers.
  • Decide the solutions using the \(\pm\) symbol. It signifies two possible roots: one by adding the square root to \(-b\), and one by subtracting it.
In the exercise, since the discriminant is negative \(-11\), the quadratic formula reveals no real solutions, only complex solutions, which are not typical when working with purely real numbers on the coordinate plane.

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