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Right triangle with the base labeled: a, the height labeled: b, and the hypotenuse labeled: c

Finding the length of the missing side of a right triangle

Find the length of the missing side of the right triangle in [link] .

Right triangle with the base labeled: a, the height labeled: 4, and the hypotenuse labeled 12.

As we have measurements for side b and the hypotenuse, the missing side is a.

a 2 + b 2 = c 2 a 2 + ( 4 ) 2 = ( 12 ) 2 a 2 + 16 = 144 a 2 = 128 a = 128 = 8 2
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Use the Pythagorean Theorem to solve the right triangle problem: Leg a measures 4 units, leg b measures 3 units. Find the length of the hypotenuse.

5 units

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Key equations

quadratic formula x = b ± b 2 4 a c 2 a

Key concepts

  • Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the equation is a difference of squares. The zero-factor property is then used to find solutions. See [link] , [link] , and [link] .
  • Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. See [link] and [link] .
  • Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. See [link] and [link] .
  • Completing the square is a method of solving quadratic equations when the equation cannot be factored. See [link] .
  • A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See [link] .
  • The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. See [link] .
  • The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See [link] .

Section exercises

Verbal

How do we recognize when an equation is quadratic?

It is a second-degree equation (the highest variable exponent is 2).

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When we solve a quadratic equation, how many solutions should we always start out seeking? Explain why when solving a quadratic equation in the form a x 2 + b x + c = 0 we may graph the equation y = a x 2 + b x + c and have no zeroes ( x -intercepts).

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When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?

We want to take advantage of the zero property of multiplication in the fact that if a b = 0 then it must follow that each factor separately offers a solution to the product being zero: a = 0   o r  b = 0.

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In the quadratic formula, what is the name of the expression under the radical sign b 2 4 a c , and how does it determine the number of and nature of our solutions?

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Describe two scenarios where using the square root property to solve a quadratic equation would be the most efficient method.

One, when no linear term is present (no x term), such as x 2 = 16. Two, when the equation is already in the form ( a x + b ) 2 = d .

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Questions & Answers

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is the branch of biology that deals with the study of microorganisms.
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What is microbiology
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studies of microbes
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How bacteria create energy to survive?
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Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
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But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
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the significance of food webs for disease transmission
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food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
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Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
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This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
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Examples of thermophilic organisms
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Give Examples of thermophilic organisms
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Prevent foreign microbes to the host
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they provide healthier benefits to their hosts
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They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
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cell is the smallest unit of life
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cell is the structural and functional unit of life
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is the fundamental units of Life
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Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
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part of a tissue or an organ being wounded or bruised.
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Binomial nomenclature
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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