/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 85 Solve each equation. (All soluti... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 85

Solve each equation. (All solutions for these equations are nonreal complex numbers.) \(m^{2}+4 m+13=0\)

Short Answer

Expert verified
The solutions are \(m = -2 + 3i\) and \(m = -2 - 3i\).

Step by step solution

01

- Identify coefficients

Identify the coefficients from the quadratic equation: Given equation: \[m^2 + 4m + 13 = 0\] Here, the coefficients are: \(a = 1\), \(b = 4\), and \(c = 13\).
02

- Apply the quadratic formula

The quadratic formula to solve \(ax^2 + bx + c = 0\) is given by: \[m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Substitute the values of \(a\), \(b\), and \(c\) into the formula: \[m = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}\]
03

- Simplify under the square root

Calculate the value under the square root: \[4^2 - 4 \cdot 1 \cdot 13 = 16 - 52 = -36\] So, the equation becomes: \[m = \frac{-4 \pm \sqrt{-36}}{2} \]
04

- Extract real and imaginary parts

Since the square root of \(-36\) is \(6i\) (where \(i\) is the imaginary unit), the equation becomes: \[m = \frac{-4 \pm 6i}{2}\] Simplify the division: \[m = -2 \pm 3i\]
05

- State the solutions

Thus, the solutions to the equation \(m^2 + 4m + 13 = 0\) are: \[m = -2 + 3i \] and \[m = -2 - 3i\]

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Key Concepts

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

quadratic formula
To solve any quadratic equation, the quadratic formula is highly effective. Given a standard quadratic equation in the form \(ax^2 + bx + c = 0\), the quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

Here鈥檚 how it works:
  • First, identify the coefficients \(a\), \(b\), and \(c\) from the equation.
  • Next, substitute these coefficients into the quadratic formula.
  • Finally, solve the expression by simplifying the terms under the square root, and then the entire fraction.

In the given exercise, \m^2 + 4m + 13 = 0\, coefficients are \(a = 1\), \(b = 4\), and \(c = 13\). When you substitute these into the quadratic formula, you follow step-by-step calculations to find the values of \(m\). Watch for negative values under the square root, as they introduce complex numbers.
complex numbers
Complex numbers extend the idea of one-dimensional number lines to two dimensions. They are written in the form \(a + bi\) where:
  • \(a\) is the real part.
  • \(b\) is the imaginary part.

For example, in the equation solution \m = -2 \pm 3i\, \(-2\) is the real part and \(3i\) is the imaginary part. Complex numbers are especially useful in cases where quadratic equations have no real solutions (as discovered when we found the square root of -36). Using complex numbers, we can express this as \(6i\), making sure we never get stuck with negative square roots.
imaginary unit
The imaginary unit, usually denoted by \(i\), is a special number with the property that \(i^2 = -1\).

Here鈥檚 a deeper insight:
  • \(i\) is not a 'real' number, but it extends our number system to include solutions to equations like \(x^2 + 1 = 0\).
  • When you take the square root of a negative number, you involve \(i\). For example, \sqrt{-36} = 6i\.

In the given solution, when simplifying \sqrt{-36}\, you extract the imaginary unit and get \6i\. This helps complete the solution of the quadratic equation and express the results as complex numbers, like \m = -2 \pm 3i.\

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Most popular questions from this chapter

Solve each equation. Check the solutions. $$9 t^{4 / 3}-25 t^{2 / 3}+16=0$$

William Froude was a l9th century naval architect who used the following expression, known as the Froude number, in shipbuilding. $$ \frac{v^{2}}{g \ell} $$ This expression was also used by \(R .\) McNeill Alexander in his research on dinosaurs. (Data from "How Dinosaurs Ran," Scientific American.) Use this expression to find the value of \(v(\) in meters per second), given \(g=9.8 \mathrm{~m}\) per sec \(^{2}\). Rhinoceros: \(\ell=1.2\) Froude number \(=2.57\)

Solve each problem. When appropriate, round answers to the nearest tenth. An object is projected directly upward from the ground. After \(t\) seconds its distance in feet above the ground is $$ s(t)=144 t-16 t^{2} $$ After how many seconds will the object be \(128 \mathrm{ft}\) above the ground? (Hint: Look for a common factor before solving the equation.)

In the 1939 classic movie The Wizard of Oz, Ray Bolger's character, the Scarecrow, wants a brain. When the Wizard grants him his "Th.D." (Doctor of Thinkology), the Scarecrow replies with the following statement. Scarecrow: The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. His statement sounds like the formula for the Pythagorean theorem. To what kind of triangle does the Scarecrow refer in his statement? To what kind of triangle does the Pythagorean theorem actually refer?

The following exercises are not grouped by type. Solve each equation. $$x^{6}-10 x^{3}=-9$$
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