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Chapter 8: Problem 8

Solve. (Find all complex-number solutions.) \(3 x^{2}-7 x+2=0\)

Short Answer

Expert verified
The solutions are 2 and \(\frac{1}{3}\).

Step by step solution

01

Identify coefficients

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

Write down the quadratic formula

The quadratic formula is \(x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a}\).
03

Substitute coefficients into the quadratic formula

Plugging in the values \(a = 3\), \(b = -7\), and \(c = 2\) into the quadratic formula, we get: \[x = \frac{{-(-7) \pm \sqrt{(-7)^2 - 4(3)(2)}}}{2(3)}\] Simplify to get: \[x = \frac{{7 \pm \sqrt{49 - 24}}}{6}\] \[x = \frac{{7 \pm \sqrt{25}}}{6}\]
04

Solve for the roots

\(\sqrt{25} = 5\), so we can write: \[x = \frac{{7 \pm 5}}{6}\] This gives us two solutions: \[x = \frac{7 + 5}{6} = 2\] and \[x = \frac{7 - 5}{6} = \frac{1}{3}\]
05

Verify if solutions are complex

Through our computations, we observe that both solutions are real numbers (2 and \(\frac{1}{3}\)). Hence, there are no complex-number solutions.

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Key Concepts

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

complex-number solutions
Quadratic equations can have different types of solutions. Sometimes the solutions are real numbers, and other times they are complex numbers. A complex number is made up of a real part and an imaginary part, and it typically looks like this: \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. Here, \(i\) is the imaginary unit, which is defined as the square root of -1.
quadratic formula
The quadratic formula is a powerful tool for solving quadratic equations. It is given by the formula: \[x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a}\]. This formula gives us the solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\). The quadratic formula requires three coefficients from the equation: \(a\), \(b\), and \(c\). These coefficients are crucial because they help us calculate both the number and type of solutions the equation has. By substituting the coefficients into the formula and simplifying, we can find the values of \(x\) that solve the equation.
real roots
Roots of a quadratic equation can either be real or complex. Real roots occur when the discriminant (the part of the quadratic formula under the square root, \(b^2 - 4ac\)) is greater than or equal to zero. If the discriminant is positive, there are two distinct real roots. If it's zero, there's exactly one real root (also known as a repeated root). If the discriminant is negative, then the roots are complex and involve imaginary numbers. In the exercise, we found that the discriminant was 25, which is a positive real number. So, the equation has two real roots: \(2\) and \(\frac{1}{3}\).

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