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Chapter 1: Problem 77

Quadratic Equations Find all real solutions of the quadratic equation. $$7 x^{2}-2 x+4=0$$

Short Answer

Expert verified
The quadratic equation has no real solutions.

Step by step solution

01

Write down the quadratic equation

The given quadratic equation is \( 7x^2 - 2x + 4 = 0 \). We need to find the real solutions for \( x \). A quadratic equation is typically of the form \( ax^2 + bx + c = 0 \), where \( a = 7 \), \( b = -2 \), and \( c = 4 \) in our example.
02

Calculate the discriminant

The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( D = b^2 - 4ac \). Substitute the values of \( a \), \( b \), and \( c \) into the formula: \[ D = (-2)^2 - 4 \times 7 \times 4 = 4 - 112 = -108. \] Since \( D < 0 \), this indicates 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
The discriminant is an important part of solving quadratic equations, helping us understand the nature of the solutions. It is represented by the symbol \( D \) and comes from the formula \( D = b^2 - 4ac \). This formula is derived from the coefficients \( a \), \( b \), and \( c \) in the standard form of a quadratic equation \( ax^2 + bx + c = 0 \).
  • If \( D > 0 \): The quadratic equation has two distinct real solutions.
  • If \( D = 0 \): The equation has exactly one real solution, which is repeated (a double root).
  • If \( D < 0 \): There are no real solutions, as the roots are complex (non-real).
In our exercise, we found that \( D = -108 \), which is less than zero. Therefore, the equation \( 7x^2 - 2x + 4 = 0 \) does not have any real solutions.
Real Solutions
In solving quadratic equations, finding the real solutions involves determining the values of \( x \) that satisfy the equation. Real solutions refer to the actual number values that can plug into the original equation to make it true.
However, not all quadratic equations will have real solutions. The presence and number of real solutions depend heavily on the discriminant \( D \).
  • If the discriminant is positive, the equation will have two real solutions.
  • If the discriminant is zero, there will be exactly one real solution.
  • When the discriminant is negative (as in our exercise), there are no real solutions.
In cases where \( D < 0 \), like with \( 7x^2 - 2x + 4 = 0 \), the equation has solutions in the set of complex numbers, but not in the real number system.
Quadratic Formula
The quadratic formula provides a foolproof way of finding the solutions to any quadratic equation. It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]This formula calculates the roots of the quadratic equation \( ax^2 + bx + c = 0 \) using the coefficients \( a \), \( b \), and \( c \).
  • The part \( \pm \sqrt{b^2 - 4ac} \) indicates that there can be two solutions: one for the plus sign and one for the minus sign.
  • The discriminant \( b^2 - 4ac \) inside the square root tells us about the nature of these solutions.
  • If it turns out to be positive, our solutions are real and distinct.
  • With a zero determinant, we have precisely one solution because the plus and minus terms cancel out.
  • When negative, as in our exercise, the expression under the square root is not real, thus leading to complex solutions.
In conclusion, while the quadratic formula is incredibly robust, its applicability to real number solutions is conditioned by the value of the discriminant. Therefore, understanding and calculating the discriminant is crucial before applying this formula.

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Most popular questions from this chapter

Solve the equation for the variable \(x\). The constants \(a\) and \(b\) represent positive real numbers. $$x^{4}-5 a x^{2}+4 a^{2}=0$$

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