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Chapter 9: Problem 68

Solve each equation. Write all solutions in bi or a \(+\) bi form. $$x^{2}+3 x+3=0$$

Short Answer

Expert verified
The solutions to the equation are \( x = -\frac{3}{2} + \frac{\sqrt{3}}{2} i \) and \( x = -\frac{3}{2} - \frac{\sqrt{3}}{2} i \)

Step by step solution

01

Identify a,b and c

In given quadratic equation \( x^{2}+3x+3=0 \), we compare it with standard form \( ax^{2}+bx+c=0 \) and get \( a=1, b=3, c=3 \)
02

Calculate the discriminant D

The discriminant \( D \) is given by \( D = b^{2} - 4ac \). Smear in the values \( a=1, b=3, c=3 \) into it to get \( D = (3^{2}) - 4.1.3 = 9 - 12 = -3 \)
03

Use the quadratic formula

The roots \( x \) given by quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \) are \( x=\frac{-3\pm \sqrt{-3}}{2}= \frac{-3}{2}\pm \frac{\sqrt{3}}{2} i \)

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Key Concepts

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

Quadratic Equations
Quadratic equations are fundamental in algebra and appear in the form of \( ax^2 + bx + c = 0 \). They are called "quadratic" because "quad" emphasizes the squared term, which is the highest power of the variable \( x \) in the equation. In our example, \( a = 1 \), \( b = 3 \), and \( c = 3 \).
The solutions to these equations are commonly referred to as "roots." These roots can be real or complex numbers, depending on the discriminant. Solving quadratic equations is essential since these equations model various real-world phenomena.
  • Solve for \( x \) by factoring.
  • Use the quadratic formula: \( x = \frac{-b \pm \sqrt{D}}{2a} \).
  • Graphically, it shows where the parabola intercepts the \( x \)-axis.
Discriminant
The discriminant in a quadratic equation, denoted as \( D \), helps determine the nature of the roots of the quadratic equation. It is calculated using the formula \( D = b^2 - 4ac \). In our problem, the calculated discriminant is \( -3 \).
The discriminant provides insight into:
  • If \( D > 0 \), two distinct real roots exist.
  • If \( D = 0 \), there is one real repeated root.
  • If \( D < 0 \), as in our case, the quadratic equation has two complex roots.
Complex solutions arise when the discriminant is negative, indicating that the roots will involve imaginary numbers.
Quadratic Formula
The quadratic formula is a universal method for solving any quadratic equation, \( ax^2 + bx + c = 0 \). The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which provides two solutions by interpreting the "plus-minus" sign (\( \pm \)) as a way to find both roots.
By substituting the values \( a = 1 \), \( b = 3 \), and \( c = 3 \), we calculate:
  • Discriminant \( D = -3 \) already computed
  • Roots: \( x = \frac{-3 \pm \sqrt{-3}}{2} \)
This formula is essential because it is effective regardless of the factorability of the quadratic equation. It simplifies directly to the complex solutions due to the negative discriminant discussed earlier.
Imaginary Numbers
Imaginary numbers play a crucial role when dealing with square roots of negative numbers, a situation encountered when the discriminant of a quadratic is negative. The main component is \( i \), defined by the property \( i^2 = -1 \).
In our solution, \( \sqrt{-3} \) becomes \( \sqrt{3}i \) and indicates complex roots. These are expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers.
  • The real part: \( -\frac{3}{2} \).
  • The imaginary part: \( \frac{\sqrt{3}}{2}i \).
Imaginary and complex numbers expand the types of solutions available beyond what real numbers can offer, providing deeper insights into mathematical problems and their applications.

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Most popular questions from this chapter

Solve each equation. Give both the exact answer and a decimal approximation to the nearest tenth. $$3 x^{2}=6 x+2$$

Solve each equation. Write the answer in bi or a \(+\) bi form. $$2 x^{2}-3 x+2=0$$

Solve each equation. Write the answer in bi or a \(+\) bi form. $$3 x^{2}-4 x+2=0$$

Solve each equation. Give both the exact answer and a decimal approximation to the nearest tenth. $$6 x^{2}-8 x+1=0$$

Simplify. Write each result in a + bi form. $$(4-i)(2+i)$$
See all solutions

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