Misplaced Pages

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57-411: The book was written completely in verse and does not contain any kind of mathematical notation. Nevertheless, it contained the first clear description of the quadratic formula (the solution of the quadratic equation). Brāhmasphuṭasiddhānta is one of the first books to provide concrete ideas on positive numbers , negative numbers , and zero. For example, it notes that the sum of a positive number and

114-395: A {\displaystyle x=u-{\tfrac {b}{2a}}} ⁠ into ⁠ a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} ⁠ , expanding the products and combining like terms, and then solving for ⁠ u 2 {\displaystyle \textstyle u^{2}\!} ⁠ , we have:

171-408: A ) 2 = b 2 − 4 a c 4 a 2 . {\displaystyle {\begin{aligned}x^{2}+2\left({\frac {b}{2a}}\right)x+\left({\frac {b}{2a}}\right)^{2}&=-{\frac {c}{a}}+\left({\frac {b}{2a}}\right)^{2}\\[5mu]\left(x+{\frac {b}{2a}}\right)^{2}&={\frac {b^{2}-4ac}{4a^{2}}}.\end{aligned}}} Because

228-451: A x + c 2 a = 0 {\displaystyle \textstyle {\tfrac {1}{2}}x^{2}+{\tfrac {b}{2a}}x+{\tfrac {c}{2a}}=0} ⁠ , then substituting the new coefficients into the standard quadratic formula. Because this variant allows re-use of the intermediately calculated quantity ⁠ b 2 a {\displaystyle {\tfrac {b}{2a}}} ⁠ , it can slightly reduce

285-397: A ± b 2 − 4 a c 2 a , β = 1 2 ( α + β ) − 1 2 ( α − β ) = − b 2 a ∓ b 2 − 4 a c 2

342-405: A ) 2 − c a , {\displaystyle x=-{\frac {b}{2a}}\pm {\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {c}{a}}}},} which can be derived by first dividing a quadratic equation by ⁠ 2 a {\displaystyle 2a} ⁠ , resulting in ⁠ 1 2 x 2 + b 2

399-526: A ) 2 − 4 c a = ± b 2 − 4 a c a . {\displaystyle \alpha -\beta =\pm {\sqrt {\left(-{\frac {b}{a}}\right)^{2}-4{\frac {c}{a}}}}=\pm {\frac {\sqrt {b^{2}-4ac}}{a}}.} Therefore, α = 1 2 ( α + β ) + 1 2 ( α − β ) = − b 2

456-542: A . {\displaystyle {\begin{aligned}\alpha &={\tfrac {1}{2}}(\alpha +\beta )+{\tfrac {1}{2}}(\alpha -\beta )=-{\frac {b}{2a}}\pm {\frac {\sqrt {b^{2}-4ac}}{2a}},\\[10mu]\beta &={\tfrac {1}{2}}(\alpha +\beta )-{\tfrac {1}{2}}(\alpha -\beta )=-{\frac {b}{2a}}\mp {\frac {\sqrt {b^{2}-4ac}}{2a}}.\end{aligned}}} The two possibilities for each of ⁠ α {\displaystyle \alpha } ⁠ and ⁠ β {\displaystyle \beta } ⁠ are

513-422: A ( u − b 2 a ) 2 + b ( u − b 2 a ) + c = 0 a ( u 2 − b a u + b 2 4 a 2 ) + b ( u − b 2 a ) + c = 0

570-403: A {\displaystyle 4a} ⁠ instead to produce ⁠ ( 2 a x ) 2 {\displaystyle \textstyle (2ax)^{2}\!} ⁠ , which allows us to complete the square without need for fractions. Then the steps of the derivation are: Applying this method to a generic quadratic equation with symbolic coefficients yields the quadratic formula:

627-419: A {\displaystyle a} ⁠ , ⁠ b {\displaystyle b} ⁠ , and ⁠ c {\displaystyle c} ⁠ representing known real or complex numbers with ⁠ a ≠ 0 {\displaystyle a\neq 0} ⁠ , the values of ⁠ x {\displaystyle x} ⁠ satisfying the equation, called

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684-613: A {\displaystyle c/a} ⁠ to isolate it on the right-hand side: a x 2 | + b x + c = 0 x 2 + b a x + c a = 0 x 2 + b a x = − c a . {\displaystyle {\begin{aligned}ax^{2{\vphantom {|}}}+bx+c&=0\\[3mu]x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}&=0\\[3mu]x^{2}+{\frac {b}{a}}x&=-{\frac {c}{a}}.\end{aligned}}} The left-hand side

741-498: A {\displaystyle x_{1}x_{2}=c/a} ⁠ , one of Vieta's formulas . Alternately, it can be derived by dividing each side of the equation ⁠ a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} ⁠ by ⁠ x 2 {\displaystyle \textstyle x^{2}} ⁠ to get ⁠ c x − 2 + b x − 1 +

798-456: A ) 2 {\displaystyle \textstyle (b/2a)^{2}} ⁠ to both sides so that the left-hand side can be factored (see the figure): x 2 + 2 ( b 2 a ) x + ( b 2 a ) 2 = − c a + ( b 2 a ) 2 ( x + b 2

855-458: A 2 . {\displaystyle {\begin{aligned}a\left(u-{\frac {b}{2a}}\right)^{2}+b\left(u-{\frac {b}{2a}}\right)+c&=0\\[5mu]a\left(u^{2}-{\frac {b}{a}}u+{\frac {b^{2}}{4a^{2}}}\right)+b\left(u-{\frac {b}{2a}}\right)+c&=0\\[5mu]au^{2}-bu+{\frac {b^{2}}{4a}}+bu-{\frac {b^{2}}{2a}}+c&=0\\[5mu]au^{2}+{\frac {4ac-b^{2}}{4a}}&=0\\[5mu]u^{2}&={\frac {b^{2}-4ac}{4a^{2}}}.\end{aligned}}} Finally, after taking

912-409: A u 2 − b u + b 2 4 a + b u − b 2 2 a + c = 0 a u 2 + 4 a c − b 2 4 a = 0 u 2 = b 2 − 4 a c 4

969-435: A x 2 + b x + c = 0 4 a 2 x 2 + 4 a b x + 4 a c = 0 4 a 2 x 2 + 4 a b x + b 2 = b 2 − 4 a c ( 2 a x + b ) 2 = b 2 − 4

1026-570: A = 0 {\displaystyle \textstyle cx^{-2}+bx^{-1}+a=0} ⁠ , applying the standard formula to find the two roots ⁠ x − 1 {\displaystyle \textstyle x^{-1}\!} ⁠ , and then taking the reciprocal to find the roots ⁠ x {\displaystyle x} ⁠ of the original equation. Any generic method or algorithm for solving quadratic equations can be applied to an equation with symbolic coefficients and used to derive some closed-form expression equivalent to

1083-422: A c . {\displaystyle x_{1}={\frac {2c}{-b-{\sqrt {b^{2}-4ac}}}},\qquad x_{2}={\frac {2c}{-b+{\sqrt {b^{2}-4ac}}}}.} This variant has been jokingly called the "citardauq" formula ("quadratic" spelled backwards). When ⁠ − b {\displaystyle -b} ⁠ has the opposite sign as either ⁠ + b 2 − 4

1140-409: A c {\displaystyle \textstyle +{\sqrt {b^{2}-4ac}}} ⁠ or ⁠ − b 2 − 4 a c {\displaystyle \textstyle -{\sqrt {b^{2}-4ac}}} ⁠ , subtraction can cause catastrophic cancellation , resulting in poor accuracy in numerical calculations; choosing between the version of the quadratic formula with

1197-563: A c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.} The following method was used by many historical mathematicians: Let the roots of the quadratic equation ⁠ a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} ⁠ be ⁠ α {\displaystyle \alpha } ⁠ and ⁠ β {\displaystyle \beta } ⁠ . The derivation starts from an identity for

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1254-635: A c 2 a x + b = ± b 2 − 4 a c x = − b ± b 2 − 4 a c 2 a . ) {\displaystyle {\begin{aligned}ax^{2}+bx+c&=0\\[3mu]4a^{2}x^{2}+4abx+4ac&=0\\[3mu]4a^{2}x^{2}+4abx+b^{2}&=b^{2}-4ac\\[3mu](2ax+b)^{2}&=b^{2}-4ac\\[3mu]2ax+b&=\pm {\sqrt {b^{2}-4ac}}\\[5mu]x&={\dfrac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.{\vphantom {\bigg )}}\end{aligned}}} This method for completing

1311-469: A c {\displaystyle \textstyle \Delta =b^{2}-4ac} ⁠ is known as the discriminant of the quadratic equation. If the coefficients ⁠ a {\displaystyle a} ⁠ , ⁠ b {\displaystyle b} ⁠ , and ⁠ c {\displaystyle c} ⁠ are real numbers then when ⁠ Δ > 0 {\displaystyle \Delta >0} ⁠ ,

1368-449: A negative number is their difference or, if they are equal, zero; that subtracting a negative number is equivalent to adding a positive number; that the product of two negative numbers is positive. Some of the notions of fractions differ from the modern rational number system. For example, Brahmagupta allows division by zero resulting in a fraction with a 0 in the denominator, and defines 0/0 = 0 . In modern mathematics, division by zero

1425-432: A probability of being within the stated interval, usually corresponding to either 1 or 2  standard deviations (a probability of 68.3% or 95.4% in a normal distribution ). Operations involving uncertain values should always try to preserve the uncertainty, in order to avoid propagation of error . If n = a ± b , any operation of the form m = f ( n ) must return a value of the form m = c ± d , where c

1482-423: A square root of both sides and substituting the resulting expression for ⁠ u {\displaystyle u} ⁠ back into ⁠ x = u − b 2 a , {\displaystyle x=u-{\tfrac {b}{2a}},} ⁠ the familiar quadratic formula emerges: x = − b ± b 2 − 4

1539-439: A value of +1 or −1 separately for each, or some appropriate relation, like s 3 = s 1 · ( s 2 ) or similar. The use of ± for an approximation is most commonly encountered in presenting the numerical value of a quantity, together with its tolerance or its statistical margin of error . For example, 5.7 ± 0.2 may be anywhere in the range from 5.5 to 5.9 inclusive. In scientific usage, it sometimes refers to

1596-438: Is f ( a ) and d is the range b updated using interval arithmetic . The symbols ± and ∓ are used in chess annotation to denote a moderate but significant advantage for White and Black, respectively. Weaker and stronger advantages are denoted by ⩲ and ⩱ for only a slight advantage, and +– and –+ for a strong, potentially winning advantage, again for White and Black respectively. The plus–minus sign resembles

1653-551: Is a symbol with multiple meanings. Other meanings occur in other fields, including medicine, engineering, chemistry, electronics, linguistics, and philosophy. A version of the sign, including also the French word ou ("or"), was used in its mathematical meaning by Albert Girard in 1626, and the sign in its modern form was used as early as 1631, in William Oughtred 's Clavis Mathematicae . In mathematical formulas ,

1710-452: Is an early part of Galois theory . This method can be generalized to give the roots of cubic polynomials and quartic polynomials , and leads to Galois theory, which allows one to understand the solution of algebraic equations of any degree in terms of the symmetry group of their roots, the Galois group . This approach focuses on the roots themselves rather than algebraically rearranging

1767-666: Is now of the form ⁠ x 2 + 2 k x {\displaystyle \textstyle x^{2}+2kx} ⁠ , and we can "complete the square" by adding a constant ⁠ k 2 {\displaystyle \textstyle k^{2}} ⁠ to obtain a squared binomial ⁠ x 2 + 2 k x + k 2 = {\displaystyle \textstyle x^{2}+2kx+k^{2}={}} ⁠ ​ ⁠ ( x + k ) 2 {\displaystyle \textstyle (x+k)^{2}} ⁠ . In this example we add ⁠ ( b / 2

Brāhmasphuṭasiddhānta - Misplaced Pages Continue

1824-481: Is undefined for any field . Ashadhara, the son of Rihluka, wrote Graha-jnana with tables based on Brahma-sphuta-siddhanta in 1132. This work is also known by the names Graha-ganita , Brahma-tulyanayana , Bhaumadi-panchagraha-nayana , Kshanika-grahanayana , or simply Ashadhara . Harihara wrote an extended version of the Graha-jnana around 1575 CE. Quadratic formula In elementary algebra ,

1881-399: The roots or zeros , can be found using the quadratic formula, x = − b ± b 2 − 4 a c 2 a , {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},} where the plus–minus symbol " ⁠ ± {\displaystyle \pm } ⁠ " indicates that

1938-481: The Taylor series of the sine function: Here, the plus-or-minus sign indicates that the term may be added or subtracted depending on whether n is odd or even; a rule which can be deduced from the first few terms. A more rigorous presentation would multiply each term by a factor of (−1) , which gives +1 when n is even, and −1 when n is odd. In older texts one occasionally finds (−) , which means

1995-468: The minus–plus symbol " ⁠ ∓ {\displaystyle \mp } ⁠ " indicates that the two roots of the quadratic equation, in the same order as the standard quadratic formula, are x 1 = 2 c − b − b 2 − 4 a c , x 2 = 2 c − b + b 2 − 4

2052-519: The quadratic formula is a closed-form expression describing the solutions of a quadratic equation . Other ways of solving quadratic equations, such as completing the square , yield the same solutions. Given a general quadratic equation of the form ⁠ a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} ⁠ , with ⁠ x {\displaystyle x} ⁠ representing an unknown, and coefficients ⁠

2109-406: The ± symbol may be used to indicate a symbol that may be replaced by either of the plus and minus signs , + or − , allowing the formula to represent two values or two equations. If x = 9 , one may give the solution as x = ±3 . This indicates that the equation has two solutions: x = +3 and x = −3 . A common use of this notation is found in the quadratic formula which describes

2166-494: The arithmetic involved. A lesser known quadratic formula, first mentioned by Giulio Fagnano , describes the same roots via an equation with the square root in the denominator (assuming ⁠ c ≠ 0 {\displaystyle c\neq 0} ⁠ ): x = 2 c − b ∓ b 2 − 4 a c . {\displaystyle x={\frac {2c}{-b\mp {\sqrt {b^{2}-4ac}}}}.} Here

2223-697: The coefficient ⁠ a ≠ 0 {\displaystyle a\neq 0} ⁠ , we can divide the quadratic equation by ⁠ a {\displaystyle a} ⁠ to obtain a monic polynomial with the same roots. Namely, x 2 + b a x + c a = ( x − α ) ( x − β ) = x 2 − ( α + β ) x + α β . {\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=(x-\alpha )(x-\beta )=x^{2}-(\alpha +\beta )x+\alpha \beta .} This implies that

2280-433: The coefficients; take the square root of both sides; and then isolate ⁠ x {\displaystyle x} ⁠ . We start by dividing the equation by the quadratic coefficient ⁠ a {\displaystyle a} ⁠ , which is allowed because ⁠ a {\displaystyle a} ⁠ is non-zero. Afterwards, we subtract the constant term ⁠ c /

2337-416: The equation has two distinct real roots; when ⁠ Δ = 0 {\displaystyle \Delta =0} ⁠ , the equation has one repeated real root; and when ⁠ Δ < 0 {\displaystyle \Delta <0} ⁠ , the equation has no real roots but has two distinct complex roots, which are complex conjugates of each other. Geometrically,

Brāhmasphuṭasiddhānta - Misplaced Pages Continue

2394-553: The equation has two roots. Written separately, these are: x 1 = − b + b 2 − 4 a c 2 a , x 2 = − b − b 2 − 4 a c 2 a . {\displaystyle x_{1}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}},\qquad x_{2}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}.} The quantity ⁠ Δ = b 2 − 4

2451-525: The equation takes the form ⁠ u 2 = s {\displaystyle \textstyle u^{2}=s} ⁠ in terms of a new variable ⁠ u {\displaystyle u} ⁠ and some constant expression ⁠ s {\displaystyle s} ⁠ , whose roots are then ⁠ u = ± s {\displaystyle u=\pm {\sqrt {s}}} ⁠ . By substituting ⁠ x = u − b 2

2508-526: The left-hand side is now a perfect square, we can easily take the square root of both sides: x + b 2 a = ± b 2 − 4 a c 2 a . {\displaystyle x+{\frac {b}{2a}}=\pm {\frac {\sqrt {b^{2}-4ac}}{2a}}.} Finally, subtracting ⁠ b / 2 a {\displaystyle b/2a} ⁠ from both sides to isolate ⁠ x {\displaystyle x} ⁠ produces

2565-406: The opposite sign to ± . The above expression can be rewritten as x ± ( y − z ) to avoid use of ∓ , but cases such as the trigonometric identity are most neatly written using the "∓" sign: which represents the two equations: Another example is the conjugate of the perfect squares which represents the two equations: A related usage is found in this presentation of the formula for

2622-1190: The original equation. Given a monic quadratic polynomial ⁠ x 2 + p x + q {\displaystyle \textstyle x^{2}+px+q} ⁠ assume that ⁠ α {\displaystyle \alpha } ⁠ and ⁠ β {\displaystyle \beta } ⁠ are the two roots. So the polynomial factors as x 2 + p x + q = ( x − α ) ( x − β ) = x 2 − ( α + β ) x + α β {\displaystyle {\begin{aligned}x^{2}+px+q&=(x-\alpha )(x-\beta )\\[3mu]&=x^{2}-(\alpha +\beta )x+\alpha \beta \end{aligned}}} which implies ⁠ p = − ( α + β ) {\displaystyle p=-(\alpha +\beta )} ⁠ and ⁠ q = α β {\displaystyle q=\alpha \beta } ⁠ . Plus%E2%80%93minus sign The plus–minus sign or plus-or-minus sign , ± ,

2679-669: The parabola's axis of symmetry . The standard way to derive the quadratic formula is to apply the method of completing the square to the generic quadratic equation ⁠ a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} ⁠ . The idea is to manipulate the equation into the form ⁠ ( x + k ) 2 = s {\displaystyle \textstyle (x+k)^{2}=s} ⁠ for some expressions ⁠ k {\displaystyle k} ⁠ and ⁠ s {\displaystyle s} ⁠ written in terms of

2736-419: The quadratic formula. Alternative methods are sometimes simpler than completing the square, and may offer interesting insight into other areas of mathematics. Instead of dividing by ⁠ a {\displaystyle a} ⁠ to isolate ⁠ x 2 {\displaystyle \textstyle x^{2}\!} ⁠ , it can be slightly simpler to multiply by ⁠ 4

2793-438: The quadratic formula: x = − b ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.} The quadratic formula can equivalently be written using various alternative expressions, for instance x = − b 2 a ± ( b 2

2850-516: The roots represent the ⁠ x {\displaystyle x} ⁠ values at which the graph of the quadratic function ⁠ y = a x 2 + b x + c {\displaystyle \textstyle y=ax^{2}+bx+c} ⁠ , a parabola , crosses the ⁠ x {\displaystyle x} ⁠ -axis: the graph's ⁠ x {\displaystyle x} ⁠ -intercepts. The quadratic formula can also be used to identify

2907-411: The same two roots in opposite order, so we can combine them into the standard quadratic equation: x = − b ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.} An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents , which

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2964-515: The same. When the standard presumption that the plus-or-minus signs all take on the same value of +1 or all −1 is not true, then the line of text that immediately follows the equation must contain a brief description of the actual connection, if any, most often of the form "where the ‘±’ signs are independent" or similar. If a brief, simple description is not possible, the equation must be re-written to provide clarity; e.g. by introducing variables such as s 1 , s 2 , ... and specifying

3021-411: The square is ancient and was known to the 8th–9th century Indian mathematician Śrīdhara . Compared with the modern standard method for completing the square, this alternate method avoids fractions until the last step and hence does not require a rearrangement after step 3 to obtain a common denominator in the right side. Another derivation uses a change of variables to eliminate the linear term. Then

3078-706: The square of a difference (valid for any two complex numbers), of which we can take the square root on both sides: ( α − β ) 2 = ( α + β ) 2 − 4 α β α − β = ± ( α + β ) 2 − 4 α β . {\displaystyle {\begin{aligned}(\alpha -\beta )^{2}&=(\alpha +\beta )^{2}-4\alpha \beta \\[3mu]\alpha -\beta &=\pm {\sqrt {(\alpha +\beta )^{2}-4\alpha \beta }}.\end{aligned}}} Since

3135-496: The square root in the numerator or denominator depending on the sign of ⁠ b {\displaystyle b} ⁠ can avoid this problem. See § Numerical calculation below. This version of the quadratic formula is used in Muller's method for finding the roots of general functions. It can be derived from the standard formula from the identity ⁠ x 1 x 2 = c /

3192-460: The sum ⁠ α + β = − b a {\displaystyle \alpha +\beta =-{\tfrac {b}{a}}} ⁠ and the product ⁠ α β = c a {\displaystyle \alpha \beta ={\tfrac {c}{a}}} ⁠ . Thus the identity can be rewritten: α − β = ± ( − b

3249-518: The two solutions to the quadratic equation ax + bx + c = 0. Similarly, the trigonometric identity can be interpreted as a shorthand for two equations: one with + on both sides of the equation, and one with − on both sides. The minus–plus sign , ∓ , is generally used in conjunction with the ± sign, in such expressions as x ± y ∓ z , which can be interpreted as meaning x + y − z or x − y + z (but not x + y + z or x − y − z ). The ∓ always has

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