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Chapter 9: Problem 50

Solve each quadratic equation for complex solutions by the quadratic formula. Write solutions in standard form. $$ x^{2}-4 x+5=0 $$

Short Answer

Expert verified
The solutions are \( x = 2 + i \) and \( x = 2 - i \).

Step by step solution

01

Identify coefficients

Identify the coefficients in the quadratic equation. For the equation \(x^{2}-4x+5=0\), the coefficients are: \(a = 1\), \(b = -4\), and \(c = 5\).
02

Write the quadratic formula

Recall the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
03

Calculate the discriminant

Calculate the discriminant \(b^2 - 4ac\): \((-4)^2 - 4(1)(5) = 16 - 20 = -4\).
04

Substitute values into the quadratic formula

Substitute the coefficients into the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{-4}}{2(1)} = \frac{4 \pm \sqrt{-4}}{2} = \frac{4 \pm 2i}{2} \].
05

Simplify the expression

Simplify the expression to find the roots: \[ x = 2 \pm i \].
06

Write solutions in standard form

The solutions in standard form are: \[ x = 2 + i \quad \text{and} \quad x = 2 - i \].

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Key Concepts

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

Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It works for all kinds of quadratic equations, including those with real and complex solutions.
The formula is given by:

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Most popular questions from this chapter

We can use a graphing calculator to illustrate how the graph of \(y=x^{2}\) can be transformed through arithmetic operations. In the standard viewing window of your calculator, graph the following one at a time, leaving the previous graphs on the screen as you move along. $$ Y_{1}=x^{2} \quad Y_{2}=2 x^{2} \quad Y_{3}=3 x^{2} \quad Y_{4}=4 x^{2} $$ Describe the effect the successive coefficients have on the parabola.

Solve each problem. Find two numbers whose sum is 300 and whose product is a maximum.

Solve equation by using the square root property. Simplify all radicals. \((x-7)^{2}=16\)

Simplify all radicals, and combine like terms. Express fractions in lowest terms. \(\frac{12-\sqrt{27}}{9}\)

Solve equation by using the square root property. Simplify all radicals. \(\left(\frac{1}{2} x+5\right)^{2}=12\)
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