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Chapter 17: Problem 504

Solve \(3 x^{2}-5 x+4=0\) by the quadratic formula.

Short Answer

Expert verified
The two solutions to the quadratic equation \(3x^2 - 5x + 4 = 0\) are \(x_1 = \frac{5 + i\sqrt{23}}{6}\) and \(x_2 = \frac{5 - i\sqrt{23}}{6}\).

Step by step solution

01

Identify a, b, and c

In the given quadratic equation \(3x^2 - 5x + 4 = 0\), the coefficients are as follows: - \(a = 3\) - \(b = -5\) - \(c = 4\)
02

Write down the quadratic formula

Recall the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] We will plug in values of a, b, and c to find the solutions for x.
03

Substitute the values of a, b, and c into the formula

Substitute the values of a, b, and c into the quadratic formula: \[x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 (3)(4)}}{2 (3)}\]
04

Simplify the expression

Simplify the expression in the formula: \[x = \frac{5 \pm \sqrt{25 - 48}}{6}\] \[x = \frac{5 \pm \sqrt{-23}}{6}\]
05

Write the solutions in terms of complex numbers

Since the square root of a negative number results in a complex number, we need to express the solutions with an imaginary part. The imaginary unit i is defined as \(i = \sqrt{-1}\), so we can rewrite the square root in terms of i: \[x = \frac{5 \pm i \sqrt{23}}{6}\]
06

Write the final solutions

The two solutions to the quadratic equation \(3x^2 - 5x + 4 = 0\) are given by: 1. \(x_1 = \frac{5 + i\sqrt{23}}{6}\) 2. \(x_2 = \frac{5 - i\sqrt{23}}{6}\)

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Most popular questions from this chapter

Discuss the nature of the roots of (a) \(3 x^{2}-7 x+3=0\) (b) \(5 \mathrm{x}^{2}+3 \mathrm{x}+1=0\)

Find the sum and the product of the roots in each of the following equations: \(x^{2}-3 x+2=0,2 x^{2}+8 x-5=0\), and \(\sqrt{2} \mathrm{x}^{2}+5 \mathrm{x}-\sqrt{8}=0\)

Use the quadratic formula to solve the following equation for \(x: x^{2}+2 x+4=0\)

Show that the graph of \(\mathrm{y}=-\mathrm{x}^{2}+\mathrm{x}-1\) has no real zeros.

If the equation \(\mathrm{x}^{2}+2(\mathrm{k}+2) \mathrm{x}+9 \mathrm{k}=0\) has equal roots, find \(\mathrm{k}\).
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