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Chapter 1: Problem 73

Use the discriminant to determine whether each quadratic equation has two unequal real solutions, a repeated real solution (a double root), or no real solution, without solving the equation. $$ 2 x^{2}-6 x+7=0 $$

Short Answer

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Step by step solution

01

- Identify coefficients

For the quadratic equation in the form \( ax^2 + bx + c = 0 \), identify the coefficients: \(a\), \(b\), and \(c\). Here, the equation is \( 2x^2 - 6x + 7 = 0 \). Therefore, \(a = 2\), \(b = -6\), and \(c = 7\).
02

- Write the discriminant formula

The discriminant \( \text{D} \) of a quadratic equation is given by \( \text{D} = b^2 - 4ac \). Write down this formula.
03

- Substitute the coefficients into the discriminant formula

Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \[ \text{D} = (-6)^2 - 4(2)(7) \].
04

- Calculate the discriminant

Perform the calculations to find the value of the discriminant: \[ \text{D} = 36 - 56 = -20 \].
05

- Determine the nature of the solutions

Interpret the value of the discriminant: If \( \text{D} > 0 \), there are two unequal real solutions. If \( \text{D} = 0 \), there is a repeated real solution (a double root). If \( \text{D} < 0 \), there are no real solutions. Here, \( \text{D} = -20 < 0 \), so 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.

quadratic equations
A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. The term 'quadratic' comes from 'quadratus,' the Latin word for square, as the highest power of \( x \) is 2.
Here's some more information about quadratic equations:
  • When the quadratic equation is set to zero, it is known as the standard form.
  • In the standard form \( ax^2 + bx + c = 0 \), \( a \) must not be zero because it would then become a linear equation.
  • Quadratic equations can be solved using different methods, such as factorization, completing the square, and using the quadratic formula.
The most importance concept when analyzing quadratic equations is the discriminant, which determines the nature of the roots.
nature of roots
The nature of the roots of a quadratic equation is determined by the discriminant, \( D \). The discriminant is a part of the quadratic formula used to find the solutions of a quadratic equation. It is calculated using the formula \( D = b^2 - 4ac \). This is derived from the standard form of the quadratic equation \( ax^2 + bx + c = 0 \).
The value of the discriminant indicates the nature of the roots:
  • If \( D > 0 \), the equation has two distinct real roots.
  • If \( D = 0 \), the equation has exactly one real root, also known as a double root.
  • If \( D < 0 \), the equation has no real roots but instead two complex roots.
Therefore, understanding the discriminant is essential for determining whether the solutions to a quadratic equation are real or complex.
real solutions versus complex solutions
The solutions of a quadratic equation depend on the discriminant and can be either real or complex.
Real Solutions:
Real solutions occur when the discriminant (\( D \)) is greater than or equal to zero.
  • Two Unequal Real Solutions: When \( D > 0 \), the quadratic equation has two different real roots. These can be found at the points where the parabola crosses the x-axis.
  • Repeated Real Solution: When \( D = 0 \), the quadratic equation has one repeated real solution, also called a double root. This means that the parabola touches the x-axis at exactly one point.

Complex Solutions:
Complex solutions occur when the discriminant (\( D \)) is less than zero.
  • When \( D < 0 \), the quadratic equation has no real roots. Instead, it has two complex roots, which are conjugates of each other. This means the parabola does not intersect the x-axis at any point.
To sum up, the value of the discriminant helps you quickly identify the type of solutions you can expect from a quadratic equation, whether they are real or complex.

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