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Chapter 7: Problem 22

solve the given quadratic equations, using the quadratic formula. Exercises \(5-8\) are the same as Exercises \(11-14\) of Section 7.2. $$2 d(d-2)=-7$$

Short Answer

Expert verified
The solutions are \(d = 1 \pm \frac{\sqrt{10}i}{2}\). The roots are complex numbers.

Step by step solution

01

Rewrite the equation

First, we want to express the quadratic equation in the standard form, which is \(ax^2 + bx + c = 0\). Start by expanding and rearranging the given equation \(2d(d-2) = -7\). Expand the left side to get \(2d^2 - 4d\). Then, add 7 to both sides to set the equation to zero: \(2d^2 - 4d + 7 = 0\).
02

Identify coefficients

Identify the coefficients \(a\), \(b\), and \(c\) in the quadratic equation \(2d^2 - 4d + 7 = 0\). Here, \(a = 2\), \(b = -4\), and \(c = 7\).
03

Apply the quadratic formula

The quadratic formula is given by \(d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \(d = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(7)}}{2(2)}\).
04

Calculate the discriminant

Calculate the discriminant inside the square root: \((-4)^2 - 4 \times 2 \times 7 = 16 - 56 = -40\). Since the discriminant is negative, the roots are complex numbers.
05

Simplify under the square root

Simplify \(\sqrt{-40}\). Rewrite it as \(\sqrt{40} \cdot i\), where \(i\) is the imaginary unit. Since \(40 = 4 \times 10\), find \(\sqrt{40} = 2\sqrt{10}\). Thus, \(\sqrt{-40} = 2\sqrt{10}i\).
06

Final solution with complex roots

Substitute back the simplified square root into the quadratic formula: \(d = \frac{4 \pm 2\sqrt{10}i}{4}\). Simplify the fraction: \(d = 1 \pm \frac{\sqrt{10}i}{2}\). There are two complex solutions: \(d = 1 + \frac{\sqrt{10}i}{2}\) and \(d = 1 - \frac{\sqrt{10}i}{2}\).

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Key Concepts

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

Complex Numbers
Complex numbers are numbers that have two parts: a real part and an imaginary part. They are typically written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\), which allows us to handle square roots of negative numbers. This inclusion of an imaginary part makes complex numbers a complete number system.
  • Real part: In our scenario, \(1\) is the real component.
  • Imaginary part: Here, \(\frac{\sqrt{10}i}{2}\) is the imaginary component.
  • Complex conjugate: For a complex number \(a + bi\), its complex conjugate is \(a - bi\). In our solutions, the complex conjugates are \(1 + \frac{\sqrt{10}i}{2}\) and \(1 - \frac{\sqrt{10}i}{2}\).
Complex numbers are crucial in solving quadratic equations with negative discriminants, leading us to solutions that "extend" the real number system to ensure that all quadratic equations have solutions.
Discriminant
The discriminant, denoted as \(b^2 - 4ac\), is a critical part of the quadratic formula \(d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It helps us determine the nature of the roots of a quadratic equation.
  • Positive Discriminant: When \(b^2 - 4ac > 0\), the quadratic has two distinct real roots.
  • Zero Discriminant: When \(b^2 - 4ac = 0\), it has exactly one real root (a double root).
  • Negative Discriminant: In our example, \(b^2 - 4ac\) was \(-40\), indicating two complex roots. This happens because the square root of a negative number introduces an imaginary component, leading to complex solutions.
The discriminant is a concise way to quickly assess what kind of solutions a quadratic equation will yield, without needing to solve the entire equation.
Imaginary Numbers
Imaginary numbers arise when we take the square root of a negative number. Since the square of any real number is always positive, mathematicians defined an imaginary unit \(i\) such that \(i^2 = -1\) to handle these scenarios.
  • Imaginary Unit: \(i = \sqrt{-1}\).
  • Using \(i\): When the discriminant is negative, as in our problem \(-40\), we rewrite it using \(i\). Thus, \(\sqrt{-40} = \sqrt{40}i = 2\sqrt{10}i\).
  • Operations: Imaginary numbers follow similar operation rules as real numbers, with additional rules for multiplication (\(i \times i = i^2 = -1\)).
Imaginary numbers, when combined with real numbers, form complex numbers, enabling us to solve equations that do not have real number solutions, thereby expanding our mathematical toolkit.

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Most popular questions from this chapter

Solve the given problems. All numbers are accurate to at least two significant digits. Without drawing the graph or completely solving the equation, explain how to find the number of \(x\) -intercepts of a quadratic function.

Solve the given equations involving fractions. $$\frac{1}{x-3}+\frac{4}{x}=2$$

Sketch the graph of each parabola by using the vertex, the \(y\) -intercept, and the \(x\) -intercepts. Check the graph using \(a\) calculator. \(y=x^{2}-4\)

Solve the given problems. All numbers are accurate to at least two significant digits. A student cycled \(3.0 \mathrm{km} / \mathrm{h}\) faster to college than when returning, which took 15 min longer. If the college is \(4.0 \mathrm{km}\) from home, what were the speeds to and from college?

Solve the given applied problem. The shape of the Gateway Arch in St. Louis can be approximated by the parabola \(y=192-0.0208 x^{2}\) (in meters) if the origin is at ground level, under the center of the Arch. (A precise equation is given in Exercise 46 of Section 13.1 ). Display the equation representing the Arch on a calculator. How high and wide is the Arch?
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