Merge pull request #47 from pie-framework/feat/equivalence-between-quadratic-equations

feat/equivalence-between-quadratic-equations

Author: CarlaCostea

README.md

@@ -61,6 +61,7 @@ It can determine mathematical equivalence between:
 
 - linear equations in one variable
 - linear equations in two variables
+- quadratic equations
 - 2-way inequalities in one or two variables
 - compound inequalities in one variable
 - trigonometric identities and functions

src/__tests__/utils.spec.ts

@@ -7,6 +7,7 @@ import {
   getCoefficients,
   setXToOne,
   solveLinearEquation,
+  solveQuadraticEquation,
   expressionsCanBeCompared,
   transformEqualityInExpression,
 } from "../symbolic/utils";
@@ -107,7 +108,7 @@ describe("getCoefficients", () => {
     ${"x - x - 2"}                 | ${[]}
   `("$expression => $coefficients", ({ expression, coefficients }) => {
     const equation = atm.convert(lta.convert(expression));
-    const coefficientsList = getCoefficients(equation);
+    const coefficientsList = getCoefficients(equation, false);
 
     expect(coefficientsList).toEqual(coefficients);
   });
@@ -116,21 +117,21 @@ describe("getCoefficients", () => {
 describe("getCoefficients", () => {
   it('equation: "x = x" - has coefficients [0, 0]', () => {
     const equation = atm.convert(lta.convert("x-x"));
-    const coefficientsList = getCoefficients(equation);
+    const coefficientsList = getCoefficients(equation, false);
 
     expect(coefficientsList).toEqual([0, 0]);
   });
 
   it('equation: "1 = -2" - if equation has no coefficient for x but can be rationalized it will return an empty array', () => {
     const equation = atm.convert(lta.convert("1+2"));
-    const coefficientsList = getCoefficients(equation);
+    const coefficientsList = getCoefficients(equation, false);
 
     expect(coefficientsList).toEqual([]);
   });
 
   it('equation: "m + n = - 2" - if equation has more than one variable, will return coefficients [1, 0]', () => {
     const equation = atm.convert(lta.convert("m+n = - 2"));
-    const coefficientsList = getCoefficients(equation);
+    const coefficientsList = getCoefficients(equation, false);
 
     expect(coefficientsList).toEqual([1, 0]);
   });
@@ -172,6 +173,26 @@ describe("solveLinearEquation", () => {
   });
 });
 
+describe("solveQuadraticEquation", () => {
+  it.each`
+    coefficients   | roots
+    ${[14, -9, 1]} | ${[{ re: 7, im: 0 }, { re: 2, im: 0 }]}
+    ${[-3, 2, 8]}  | ${[{ re: 0.5, im: 0 }, { re: -0.75, im: 0 }]}
+    ${[8, 6, 2]}   | ${[{ re: -1.5, im:  1.3228756555323 }, { re: -1.5, im: -1.3228756555323 }]}
+    ${[6, 5, 1]}   | ${[{ re: -2, im: 0 }, { re: -3, im: 0 }]}
+    ${[3, -7, 2]}  | ${[{ re: 3, im: 0 }, { re: 1/2, im: 0 }]}
+    ${[1, 3, 2]}   | ${[{ re: -0.5, im: 0 }, { re: -1, im: 0 }]}
+    ${[-3, 2, 1]}  | ${[{ re: 1, im: 0 }, { re: -3, im: 0 }]}
+    ${[-9, 0, 1]}  | ${[{ re: 3, im: 0 }, { re: -3, im: 0 }]}
+    ${[-8, -5, 3]} | ${[{ re: 2.6666666666667, im: 0 }, { re: -1, im: 0 }]}
+    ${[17, -5, 4]} | ${[{ re: 0.625, im: 1.9645292056877 }, { re: 0.625, im: -1.9645292056877}]}
+  `("$coefficients => $roots", ({ coefficients, roots }) => {
+    const result = solveQuadraticEquation(coefficients);
+    
+    expect(result).toEqual(roots);
+  });
+});
+
 describe("solveLinearEquation", () => {
   it('equation: "x = x" - has infinite solutions', () => {
     const coefficients = [0, 0];

src/fixtures/latex-equal/symbolic/3-way-inequalities-with-variables.ts

@@ -50,23 +50,23 @@ export default {
     },
     {
       target: "1 < 2x < 10",
-       eq: ["2 < 4x < 20"],
+      eq: ["2 < 4x < 20"],
       ne: [
-         "2 = 4x = 20", 
+        "2 = 4x = 20",
         "1 <2x >10"
       ],
     },
     {
       target: "a≠b≠c",
-       ne: ["c≠d≠e"]
+      ne: ["c≠d≠e"]
     },
     {
       target: "a=b=c",
-       ne: ["c=d=e"]
+      ne: ["c=d=e"]
     },
     {
       target: "-99 ≤  9-12r ≤  45",
-       eq: ["-108≤ -12r ≤  36", "-108/-12≤ -12r/-12 ≤  36/-12","9 ≥ r ≥  -3", "-3≤ r≤ 9", ],
+      eq: ["-108≤ -12r ≤  36", "-108/-12≤ -12r/-12 ≤  36/-12", "9 ≥ r ≥  -3", "-3≤ r≤ 9",],
       ne: [
         "-108≤ -12r < 36",
         "-108≤ -12r >  36",

src/fixtures/latex-equal/symbolic/fractions-symbolic.ts

@@ -419,5 +419,26 @@ export default {
         "60x-900=y^2-14y+49",
       ],
     },
+    {
+      target: "\\frac{6}{(x+1)^2}+\\frac{1}{x+1}",
+      eq: [
+        "\\frac{6}{(x+1)^2}+\\frac{5}{5(x+1)}",
+        "\\frac{6}{(x+1)(x+1)}+\\frac{5}{5(x+1)}",
+        "\\frac{6}{x(x+1)+1(x+1)}+\\frac{5}{5(x+1)}",
+        "\\frac{6}{x^2+x+1(x+1)}+\\frac{20}{20(x+1)}",
+      ],
+    },
+    {
+      target: "\\frac{2}{(x+5)}+\\frac{1}{x+2}+10",
+      eq: [
+        "\\frac{4}{2(x+5)}+\\frac{3}{3(x+2)}+10",
+        "\\frac{4}{2(x+5)}+\\frac{3}{3(x+2)}+5+5",
+        "\\frac{10}{5(x+5)}+\\frac{1}{1(x+4-2)}+20-10",
+      ],
+      ne: [
+        "\\frac{1}{(x+5)}+\\frac{3}{3(x+2)}+10",
+        "\\frac{2}{(x+4)}+\\frac{1}{x+2}+10",
+      ],
+    },
   ],
 };

src/fixtures/latex-equal/symbolic/math-series.ts

@@ -9,7 +9,7 @@ export default {
     {
       target: "a_{i}=1/2i[2s+(i-1)d]",
       eq: ["a_{i}=0.5i(2s+di-d)"],
-      ne: ["a_{i-1}=0.5i(2s+di-d)"],
+      ne: ["a_{i-1}=0.5i(2s+di-d)", "0.5i(2s+di-d)=a_{i-1}", "0.5i(2s+di-d)=a"],
     },
     {
       target: "a_{n}=(-1/(2i))^n",

src/fixtures/latex-equal/symbolic/quadratic-equations.ts

@@ -1,12 +1,130 @@
 export default {
   mode: "symbolic",
-
   tests: [
     {
-      // // quadratic equation
-      //only: true,
-      // target: "x^2-7x+10=0",
-      // eq: ["2x^2-14x+20=0"],
+      target: "x^2-7x+10=0",
+      eq: ["2x^2-14x+20=0"],
+    },
+    {
+      target: "5x^2+20x+32=0",
+      eq: ["10x^2+40x+64=0", "15x^2+60x+96=0"],
+    },
+    {
+      target: "x^2 + 3x +2 = 5x +10",
+      eq: [
+        "(x+2)(x-4)=0",
+        "x^2-4x+2x-8=0"
+      ],
+      ne: ["(x+1)(x-4)=0", "(x)(x-4)=0", "x^2-4x+2x-8"],
+    },
+    {
+      target: "x^2 = 5x - 6",
+      eq: ["x^2 - 5x + 6=0", "(x-3)(x-2)=0"],
+      ne: ["(x-3)(x-1)=0", "(x)(x-3)=0", "x^2-5x - 6 = 0"],
+    },
+    {
+      target: "6x^2-3x-2x +1 =0",
+      eq: ["3x(2x-1)-1(2x-1)=0", "(3x-1)(2x-1)=0"],
+      ne: ["3x(2x)-1(2x-1)=0", "(3x-1)(2x)=0"],
+    },
+    {
+      target: "6x^2+4x+2=0",
+      eq: [
+        "3x^2+2x+1=0",
+        "x^2+\\frac{2x}{3}+\\frac{1}{3}=0",
+        "x^2+\\frac{2x}{3}+\\frac{2}{6}=0",
+      ],
+      ne: [
+        "x^2+\\frac{2x}{3}+\\frac{2}{4}=0",
+        "x^2+\\frac{2x}{3}+\\frac{2}{5}=0",
+        "x^2+\\frac{2x}{3}+\\frac{2}{3}=0",
+      ],
+    },
+    {
+      target: "3x^2-18x+14=0",
+      eq: [
+        "3(x^2-6x)+14=0",
+        "3(x^2-6x +3^2-3^2)=-14",
+        "3(x^2-6x+3^2)-3(9)+14=0",
+        "3(x-3)^2-13=0",
+      ],
+    },
+    {
+      target: "(t-5)^2-9=0",
+      eq: ["(t-5)^2=9"],
+      ne: ["(t-5)^3=9", "(x-5)^2=9", "(t-4)^2=9"],
+    },
+    {
+      target: "\\frac{6}{(x+1)^2}+\\frac{2}{2x}+2=0",
+      eq: [
+        "\\frac{6}{(x+1)(x+1)}+\\frac{2}{2x}+2=0",
+        "\\frac{6}{(x+1)(x+1)}+\\frac{2}{2x}=-2",
+        "\\frac{6}{(x+1)(x+1)}+\\frac{2}{2x}+6-4=0",
+        "\\frac{6}{(x+1)(x+1)}+\\frac{2}{2x}+6=+4",
+        "\\frac{6}{(x+1)(x+1)}+\\frac{2}{2x}+6=+4",
+        "\\frac{6}{(x+1)(x+1)}+\\frac{2}{2x}+4=+6-4",
+        "\\frac{6}{(x+1)(x+1)}+\\frac{3}{3x}+2=0",
+        "\\frac{6}{x(x+1)+1(x+1)}+\\frac{5}{5(x)}+2=0",
+      ],
+    },
+    {
+      target: "\\frac{6}{(x+1)^2}+\\frac{1}{x+1}=1",
+      eq: [
+        "\\frac{6}{(x+1)^2}+\\frac{5}{5(x+1)}=1",
+        "\\frac{6}{(x+1)(x+1)}+\\frac{5}{5(x+1)}=1",
+        "\\frac{6}{x(x+1)+1(x+1)}+\\frac{5}{5(x+1)}=1",
+        "\\frac{6}{x^2+x+1(x+1)}+\\frac{5}{5(x+1)}=1",
+      ],
+    },
+    {
+      // test square props
+      target: "(u+1)^2=0",
+      eq: [
+        // expand square
+        "(u+1)(u+1)=0",
+        // distributive property
+        "u(u+1)+1(u+1)=0",
+        "u^2+u+1(u+1)=0",
+      ],
+    },
+    {
+      // test square props
+      target: "(z-1)^2=0",
+      eq: [
+        // expand square
+        "(z-1)(z-1)=0",
+        // distributive property
+        "z(z-1)-1(z-1)=0",
+        "z*z+z*(-1)-1(z-1)=0",
+        "z*z+z*(-1)-1z-1*(-1)=0",
+        "z^2-2z+1=0",
+      ],
+    },
+    {
+      // they both have complex solutions but they should not be equal
+      target: "4x^2-5x+17=0",
+      ne: ["2x^2+6x+8=0", "x^2+4x+5=0"],
+    },
+    {
+      target: "x^2+4x+5=0",
+      eq: ["2x^2+8x+10=0"],
+    },
+    {
+      // positive real roots
+      target: "x^2+6x+5=0",
+      eq: ["3x^2+18x+15=0", "x^2+6x+4=-1", "x^2+6x=-5"],
+      ne:["4x^2+18x+15=0", "3x^2+17x+15=0", "3x^2+18x+16=0", "x^3+6x+5=0"]
+    },
+    {
+      // one real root - zero
+      target: "x^2-2x+1=0",
+      eq: ["2x^2-4x+2=0"],
+    },
+    {
+      // two complex roots
+      target: "x^2-3x+10=0",
+      eq: ["2x^2-6x+20=0"],
+      ne:["x^2-4x+10=0", "x^2-3x+10=1"]
     },
   ],
 };

src/mathjs.ts

@@ -1,5 +1,6 @@
 import { all, create } from "mathjs";
 export const mathjs = create(all, { number: "Fraction" });
 export const { parse } = mathjs;
+
 // @ts-ignore
 export const replacer = mathjs.replacer;

src/symbolic/compare-compound-inequations.ts

@@ -60,8 +60,8 @@ export const getLimit = (
   const expressionR = transformEqualityInExpression(expressionsPair.right);
   const expressionL = transformEqualityInExpression(expressionsPair.left);
 
-  const xFirstInequality = solveLinearEquation(getCoefficients(expressionR));
-  const xSecondInequality = solveLinearEquation(getCoefficients(expressionL));
+  const xFirstInequality = solveLinearEquation(getCoefficients(expressionR, true));
+  const xSecondInequality = solveLinearEquation(getCoefficients(expressionL, true));
 
   if (limitType === "inferior") {
     return Math.min(xFirstInequality, xSecondInequality);