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  • ...qualities, same quantities can be added or subtracted without changing the inequality sign, much like [[equation|equations]]. However, when multiplying, dividin ...he value <math>x = \frac{3}{2}</math> satisfies the inequality because the inequality is nonstrict.
    12 KB (1,806 words) - 05:07, 19 June 2024
  • The '''trivial inequality''' is an [[inequality]] that states that the square of any real number is nonnegative. Its name c The trivial inequality is one of the most commonly used theorems in mathematics. It is very well-k
    3 KB (583 words) - 20:20, 2 August 2024
  • ...three main strategies: [[factoring]], [[completing the square]] and the [[quadratic formula]]. The purpose of factoring is to turn a general quadratic into a product of [[binomial]]s. This is easier to illustrate than to descr
    2 KB (264 words) - 11:04, 15 July 2021
  • * [[Quadratic equation]] ** [[Triangle Inequality]]
    2 KB (198 words) - 16:47, 3 November 2021
  • '''Jensen's Inequality''' is an inequality discovered by Danish mathematician Johan Jensen in 1906. ==Inequality==
    3 KB (623 words) - 12:10, 20 February 2024
  • ...tic Mean-Geometric Mean-Harmonic Mean Inequality''' (EM-AM-GM-HM), is an [[inequality]] of the [[root-mean power]], [[arithmetic mean]], [[geometric mean]], and The quadratic mean's root mean power is 2 and the arithmetic mean's root mean power is 1,
    5 KB (913 words) - 14:44, 14 August 2024
  • ...math>. This looks like a quadratic so set <math>z= x\sin x</math> and use quadratic equation on <math>9z^2 - yz + 4 = 0</math> to see that <math>z = \frac{y\pm ...(2)}</math>, with equality holding when <math>3x\sin x=2</math>. From this inequality, we can see the following:
    5 KB (824 words) - 18:34, 20 July 2024
  • ...aximum value of <math>b^2</math> (this follows directly from the [[trivial inequality]], because if <math>{x^2 \ge 0}</math> then plugging in <math>a+n</math> fo Rigorously, we need to make sure that the equality of the AM-GM inequality is possible to be obtained (in other words, <math>(81^2 - 81x^2)</math> and
    4 KB (703 words) - 22:13, 30 August 2024
  • ==Quadratic equations== A quadratic equation has a degree of two. This means that the highest exponent in the e
    5 KB (932 words) - 11:57, 26 July 2023
  • *** [[Quadratic | Quadratics]] *** [[Quadratic | Quadratics]]
    991 bytes (86 words) - 14:58, 24 August 2024
  • * [[Quadratic mean]] (also known as the root mean square) ...hain. This inequality chain is a set of special cases of the [[Power mean inequality]].
    1 KB (148 words) - 14:28, 12 November 2023
  • ...a^2 - b^2}{2a}\right)^2 </math>, so <math>x,y </math> are the roots of the quadratic <math> m^2 - \frac{a^2 + b^2}{2a}m + \left(\frac{a^2 - b^2}{2a}\right)^2 </ ...sharper bound <math>a^2 > b^2 </math>. It is clear that both roots of the quadratic must be positive if the discriminant is positive (we can see this either fr
    5 KB (916 words) - 17:15, 26 March 2024
  • ...f positive integers satisfying this equation also satisfies the [[triangle inequality]], so the solutions correspond to right triangles with integral side length which is a contradiction, since 2 is not a [[quadratic residue | square]] [[modulus | mod]] 4. Hence at least one of <math>a</mat
    4 KB (684 words) - 15:45, 1 August 2020
  • ...ht)^2 - \frac{b^2}{4a}} \le \sqrt{\frac{-b^2}{4a}}</math> by the [[Trivial Inequality]] (remember that <math>a \le 0</math>). Since <math>f</math> is continuous ...cts what the first thing that we found. Now, first consider the graph of a quadratic equation/parabola. We know that a parabola always has a vertex, and always
    9 KB (1,606 words) - 10:34, 10 July 2020
  • The '''root-mean-square''' or ''quadratic mean'' of a collection of [[real number]]s <math>x_1,\dots , x_n</math> is ...well-known [[Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality]]
    543 bytes (86 words) - 12:41, 8 May 2013
  • By the AM-GM Inequality, <math>(u-1) + \frac{4}{u-1} \ge 4</math>, meaning that <math>\frac{V_1}{V_ ...it is the reciprocal of <math>f(x)</math>). Using the vertex formula for a quadratic function, we get <math>x = -\frac{b}{2a} = \frac{1}{2}</math> , which gives
    7 KB (1,214 words) - 17:49, 29 January 2018
  • If in applying the quadratic formula to a quadratic equation Given: <math> x > 0, y > 0, x > y </math> and <math> z\not = 0 </math>. The inequality which is not always correct is:
    23 KB (3,646 words) - 20:53, 21 June 2024
  • The set of values <math>x</math> satisfying the inequality <math>|3-x|<4</math> consists of all <math>x</math> such that: \textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad \
    22 KB (3,348 words) - 11:53, 22 July 2024
  • '''Proof 1.''' By the [[triangle inequality]], we can immediately see that <math>AD \geq 20\sqrt{7}</math>. However, no ...ath>2100 = x^2+y^2-xy \iff y^2-xy+(x^2-2100) = 0</math>. Viewing this as a quadratic in <math>y</math>, the discriminant <math>\Delta</math> must satisfy <math>
    4 KB (642 words) - 15:11, 16 November 2024
  • We claim that the inequality ...>4z^2 - 16z + 25 = 4(z-4)^2 + 9 \ge 0</math>, it suffices to show that the quadratic cannot have more than one root, or the [[discriminant]] <math>\Delta \le 0<
    4 KB (703 words) - 17:40, 3 January 2019

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