Revision #1508 → #1849 · back to history
modifiedQuadratic equation (standard form)86082b0f8dcf
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| anchors | [{"section":"(Lead)","snippet":"a quadratic equation (from Latin quadratus ' square ' ) is an equation that can be rearranged in standard form"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=0\\,,}"}] | — |
modifiedCoefficients of a quadratic equation32be33f1853f
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| anchors | [{"section":"(Lead)","snippet":"The numbers a , b , and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient , the linear coefficient and the constant coefficient or free term"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=0\\,,}"}] | — |
modifiedSolutions, roots, and double root53c8458b3e8c
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| anchors | [{"section":"(Lead)","snippet":"The values of [MATH] that satisfy the equation are called solutions of the equation, and roots or zeros of the quadratic function on its left-hand side"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0}"}] | — |
modifiedNumber of roots of a quadratic1d7aa0afa948
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| anchors | [{"section":"(Lead)","snippet":"A quadratic equation always has two roots, if complex roots are included and a double root is counted for two"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0}"}] | — |
modifiedFactored form of a quadratic5f4604d99e18
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| anchors | [{"section":"(Lead)","snippet":"A quadratic equation can be factored into an equivalent equation"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0}"}] | — |
modifiedQuadratic formulabbe8ecef0098
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| anchors | [{"section":"(Lead)","snippet":"The quadratic formula"},{"type":"math_alttext","value":"{\\displaystyle x={\\frac {-b\\pm {\\sqrt {b^{2}-4ac}}}{2a}}}"}] | — |
modifiedZero Factor Property3b125f08ac82
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| mathlib.module | Mathlib.Algebra.GroupWithZero.Basic | Mathlib.Algebra.GroupWithZero.Defs |
modifiedCompleting the square algorithm8e3319c476bd
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| anchors | [{"section":"Completing the square","snippet":"The process of completing the square makes use of the algebraic identity"},{"type":"math_alttext","value":"{\\displaystyle x^{2}+2hx+h^{2}=(x+h)^{2},}"}] | — |
modifiedSolving 2x^2 + 4x - 4 = 06d0d863b8d12
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| anchors | [{"section":"Completing the square","snippet":"We illustrate use of this algorithm by solving 2 x 2 + 4 x − 4 = 0"},{"type":"math_alttext","value":"{\\displaystyle 2x^{2}+4x-4=0}"},{"type":"math_alttext","value":"{\\displaystyle \\ x^{2}+2x-2=0}"},{"type":"math_alttext","value":"{\\displaystyle \\ x^{2}+2x=2}"},{"type":"math_alttext","value":"{\\displaystyle \\ x^{2}+2x+1=2+1}"},{"type":"math_alttext","value":"{\\displaystyle \\left(x+1\\right)^{2}=3}"},{"type":"math_alttext","value":"{\\displaystyle \\ x+1=\\pm {\\sqrt {3}}}"},{"type":"math_alttext","value":"{\\displaystyle \\ x=-1\\pm {\\sqrt {3}}}"}] | — |
modifiedDerivation of quadratic formula0c60898d54b5
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| anchors | [{"section":"Quadratic formula and its derivation","snippet":"Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula"},{"type":"math_alttext","value":"{\\displaystyle \\left(x+{\\frac {b}{2a}}\\right)^{2}={\\frac {b^{2}-4ac}{4a^{2}}}.}"},{"type":"math_alttext","value":"{\\displaystyle x={\\frac {-b\\pm {\\sqrt {b^{2}-4ac}}}{2a}}.}"}] | — |
modifiedMuller's method formula7278f5245885
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| anchors | [{"section":"Quadratic formula and its derivation","snippet":"A lesser known quadratic formula, as used in Muller's method , provides the same roots via the equation"},{"type":"math_alttext","value":"{\\displaystyle x={\\frac {2c}{-b\\pm {\\sqrt {b^{2}-4ac}}}}.}"}] | — |
modifiedQuadratic formula for reduced quadraticc47439d31ffb
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| anchors | [{"section":"Reduced quadratic equation","snippet":"The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients"},{"type":"math_alttext","value":"{\\displaystyle x=-{\\frac {p}{2}}\\pm {\\sqrt {\\left({\\frac {p}{2}}\\right)^{2}-q}}\\,.}"}] | — |
modifiedDiscriminantee4dd9ce5f62
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| anchors | [{"section":"Discriminant","snippet":"the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case D or an upper case Greek delta"},{"type":"math_alttext","value":"{\\displaystyle \\Delta =b^{2}-4ac.}"}] | — |
modifiedDiscriminant determines nature of roots7359a1e99ded
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| anchors | [{"section":"Discriminant","snippet":"In this case the discriminant determines the number and nature of the roots"},{"type":"math_alttext","value":"{\\displaystyle \\Delta =b^{2}-4ac.}"}] | — |
modifiedFactor theorem for quadraticsf96fdb80769a
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| anchors | [{"section":"Quadratic factorization","snippet":"is a factor of the polynomial"},{"type":"math_alttext","value":"{\\displaystyle x-r}"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c}"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=0.}"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=a\\left(x-{\\frac {-b+{\\sqrt {b^{2}-4ac}}}{2a}}\\right)\\left(x-{\\frac {-b-{\\sqrt {b^{2}-4ac}}}{2a}}\\right).}"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=a\\left(x+{\\frac {b}{2a}}\\right)^{2}.}"}] | — |
| mathlib.module | Mathlib.Algebra.Polynomial.FieldDivision | Mathlib.Algebra.Polynomial.Div |
modifiedFactorization in repeated-root case82b5f2dd9096
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| anchors | [{"section":"Quadratic factorization","snippet":"In the special case b 2 = 4 ac where the quadratic has only one distinct root"},{"type":"math_alttext","value":"{\\displaystyle x-r}"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c}"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=0.}"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=a\\left(x-{\\frac {-b+{\\sqrt {b^{2}-4ac}}}{2a}}\\right)\\left(x-{\\frac {-b-{\\sqrt {b^{2}-4ac}}}{2a}}\\right).}"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=a\\left(x+{\\frac {b}{2a}}\\right)^{2}.}"}] | — |
modifiedGeometric expression of complex rootsd30798168030
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| anchors | [{"section":"Graphical solution","snippet":"Then the real part of the roots is h , and their imaginary part are ± d"},{"type":"math_alttext","value":"{\\displaystyle y=a(x-h)^{2}+k.}"},{"type":"math_alttext","value":"{\\displaystyle h+id\\quad {\\text{and}}\\quad h-id,}"},{"type":"math_alttext","value":"{\\displaystyle 5+3i\\quad {\\text{and}}\\quad 5-3i.}"}] | — |
modifiedBabylonian simultaneous equations problem9df2f14cf77b
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| anchors | [{"section":"History","snippet":"the problems typically involved solving a pair of simultaneous equations of the form"},{"type":"math_alttext","value":"{\\displaystyle x+y=p,\\ \\ xy=q,}"},{"type":"math_alttext","value":"{\\displaystyle z^{2}+q=pz.}"}] | — |
modifiedBrahmagupta's solutionf69c6ce78b6a
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| anchors | [{"section":"History","snippet":"the first explicit (although still not completely general) solution of the quadratic equation ax 2 + bx = c as follows"},{"type":"math_alttext","value":"{\\displaystyle x={\\frac {{\\sqrt {4ac+b^{2}}}-b}{2a}}.}"}] | — |
modifiedVieta's formulasbecf5defab72
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| anchors | [{"section":"Vieta's formulas","snippet":"Vieta's formulas (named after François Viète ) are the relations"},{"type":"math_alttext","value":"{\\displaystyle x_{1}+x_{2}=-{\\frac {b}{a}},\\quad x_{1}x_{2}={\\frac {c}{a}}}"},{"type":"math_alttext","value":"{\\displaystyle \\left(x-x_{1}\\right)\\left(x-x_{2}\\right)=x^{2}-\\left(x_{1}+x_{2}\\right)x+x_{1}x_{2}=0}"},{"type":"math_alttext","value":"{\\displaystyle x^{2}+{\\frac {b}{a}}x+{\\frac {c}{a}}=0.}"}] | — |
modifiedVertex x-coordinate as average of roots9816dc200763
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| anchors | [{"section":"Vieta's formulas","snippet":"the vertex's x -coordinate is located at the average of the roots (or intercepts)"},{"type":"math_alttext","value":"{\\displaystyle x_{V}={\\frac {x_{1}+x_{2}}{2}}=-{\\frac {b}{2a}}.}"},{"type":"math_alttext","value":"{\\displaystyle y_{V}=-{\\frac {b^{2}}{4a}}+c=-{\\frac {b^{2}-4ac}{4a}}.}"},{"type":"math_alttext","value":"{\\displaystyle ax^{2}+bx+c=a\\left(x+{\\frac {b}{2a}}\\right)^{2}-{\\frac {b^{2}-4ac}{4a}}.}"}] | — |
modifiedVieta-based root approximationa8c8d0fe48d4
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| anchors | [{"section":"Vieta's formulas","snippet":"Vieta's formulas provide a useful method for finding the roots of a quadratic equation in the case where one root is much smaller than the other"},{"type":"math_alttext","value":"{\\displaystyle x_{1}\\approx -{\\frac {b}{a}}.}"},{"type":"math_alttext","value":"{\\displaystyle x_{2}={\\frac {c}{ax_{1}}}\\approx -{\\frac {c}{b}}.}"}] | — |
modifiedWorked trigonometric solution31de6f975054
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| anchors | [{"section":"Trigonometric solution","snippet":"let us assume we had available seven-place logarithm and trigonometric tables, and wished to solve the following to six-significant-figure accuracy"},{"type":"math_alttext","value":"{\\displaystyle 4.16130x^{2}+9.15933x-11.4207=0}"},{"type":"math_alttext","value":"{\\displaystyle x_{2}=-10^{0.2192318+0.2706462}=-3.08943}"}] | — |
modifiedSquare root existence in extension fields3a06ba3db8c8
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| anchors | [{"section":"Generalization of quadratic equation","snippet":"Even if a field does not contain a square root of some number, there is always a quadratic extension field which does"},{"type":"math_alttext","value":"{\\displaystyle \\pm {\\sqrt {b^{2}-4ac}}}"}] | — |
modifiedCharacteristic 2 case with b=0a01fd65c2d2d
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| anchors | [{"section":"Characteristic 2","snippet":"If b = 0 , then the solution reduces to extracting a square root"},{"type":"math_alttext","value":"{\\displaystyle x^{2}+bx+c}"},{"type":"math_alttext","value":"{\\displaystyle x={\\sqrt {c}}}"},{"type":"math_alttext","value":"{\\displaystyle -{\\sqrt {c}}=-{\\sqrt {c}}+2{\\sqrt {c}}={\\sqrt {c}}.}"},{"type":"math_alttext","value":"{\\displaystyle \\displaystyle x^{2}+c=(x+{\\sqrt {c}})^{2}.}"}] | — |
modified2-root R(c) in characteristic 217d3b26d184f
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| anchors | [{"section":"Characteristic 2","snippet":"define the 2-root R ( c ) of c to be a root of the polynomial x 2 + x + c"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {b}{a}}R\\left({\\frac {ac}{b^{2}}}\\right)}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {b}{a}}\\left(R\\left({\\frac {ac}{b^{2}}}\\right)+1\\right).}"}] | — |
modifiedRoots of non-monic quadratic in characteristic 2d19d7ae8f287
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| anchors | [{"section":"Characteristic 2","snippet":"In terms of the 2-root operation, the two roots of the (non-monic) quadratic ax 2 + bx + c are"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {b}{a}}R\\left({\\frac {ac}{b^{2}}}\\right)}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {b}{a}}\\left(R\\left({\\frac {ac}{b^{2}}}\\right)+1\\right).}"}] | — |
addedReal coefficients give two real solutions, a double root, or complex conjugates9d6e7ca78402
addedComplex roots of real quadratic are conjugatesc6639c90668e
addedVertex y-coordinate via Vieta9740d33da26a
addedArtin–Schreier extension governs characteristic-2 quadraticsfaa9e7cda7aa