Comprehensive Guide to Quadratic Equations and Complex Numbers

Sum and Product of Roots of Quadratic Equations

  • General Formulas for Roots of Quadratic Equations:

    • For any standard quadratic equation written in the form ax2+bx+c=0ax^2 + bx + c = 0:

    • Sum of the roots=−ba\text{Sum of the roots} = -\frac{b}{a}

    • Product of the roots=ca\text{Product of the roots} = \frac{c}{a}

  • Examples of Finding Sum and Product of Roots:

    • Example 1: x2−8x+12=0x^2 - 8x + 12 = 0

    • Identify coefficients: a=1a = 1, b=−8b = -8, c=12c = 12

    • Sum=−ba=−−81=8\text{Sum} = -\frac{b}{a} = -\frac{-8}{1} = 8

    • Product=ca=121=12\text{Product} = \frac{c}{a} = \frac{12}{1} = 12

    • Example 2: 7x2+19x=67x^2 + 19x = 6

    • Rewrite in standard form: 7x2+19x−6=07x^2 + 19x - 6 = 0

    • Identify coefficients: a=7a = 7, b=19b = 19, c=−6c = -6

    • Sum=−ba=−197\text{Sum} = -\frac{b}{a} = -\frac{19}{7}

    • Product=ca=−67=−67\text{Product} = \frac{c}{a} = \frac{-6}{7} = -\frac{6}{7}

    • Example 3: x2−16=0x^2 - 16 = 0

    • Identify coefficients: a=1a = 1, b=0b = 0, c=−16c = -16

    • Sum=−ba=−01=0\text{Sum} = -\frac{b}{a} = -\frac{0}{1} = 0

    • Product=ca=−161=−16\text{Product} = \frac{c}{a} = \frac{-16}{1} = -16

    • Example 4: x2+4x=0x^2 + 4x = 0

    • Identify coefficients: a=1a = 1, b=4b = 4, c=0c = 0

    • Sum=−ba=−41=−4\text{Sum} = -\frac{b}{a} = -\frac{4}{1} = -4

    • Product=ca=01=0\text{Product} = \frac{c}{a} = \frac{0}{1} = 0

Constructing Quadratic Equations from Given Roots

  • Formula for Constructing Quadratic Equations:

    • Standard Equation Structure: x2−(sum of the roots)x+(product of the roots)=0x^2 - (\text{sum of the roots})x + (\text{product of the roots}) = 0

    • All final quadratic equation answers must have integer coefficients.

  • Constructing Equations from Sum and Product:

    • Example 1: Given Sum=−2\text{Sum} = -2 and Product=−15\text{Product} = -15

    • Formula substitution: x2−(−2)x+(−15)=0x^2 - (-2)x + (-15) = 0

    • Simplified Equation: x2+2x−15=0x^2 + 2x - 15 = 0

  • Constructing Equations from Given Roots:

    • Example 2: Roots are −7-7 and 1010

    • Sum=−7+10=3\text{Sum} = -7 + 10 = 3

    • Product=−7×10=−70\text{Product} = -7 \times 10 = -70

    • Equation: x2−3x−70=0x^2 - 3x - 70 = 0

    • Example 3: Roots are 44 and −2-2

    • Sum=4+(−2)=2\text{Sum} = 4 + (-2) = 2

    • Product=4×(−2)=−8\text{Product} = 4 \times (-2) = -8

    • Equation: x2−2x−8=0x^2 - 2x - 8 = 0

    • Example 4: Roots are 2 + \text{\sqrt{6}} and 2 - \text{\sqrt{6}}

    • \text{Sum} = (2 + \text{\sqrt{6}}) + (2 - \text{\sqrt{6}}) = 4

    • \text{Product} = (2 + \text{\sqrt{6}})(2 - \text{\sqrt{6}}) = 4 - 2\text{\sqrt{6}} + 2\text{\sqrt{6}} - 6 = -2

    • Equation: x2−4x−2=0x^2 - 4x - 2 = 0

    • Example 5: Roots are 2+i2 + i and 2−i2 - i

    • Complex roots always come in conjugate pairs.

    • Sum=(2+i)+(2−i)=4\text{Sum} = (2 + i) + (2 - i) = 4

    • Product=(2+i)(2−i)=4−2i+2i−i2=4−(−1)=5\text{Product} = (2 + i)(2 - i) = 4 - 2i + 2i - i^2 = 4 - (-1) = 5

    • Equation: x2−4x+5=0x^2 - 4x + 5 = 0

  • Solving for Unknown Roots and Parameters:

    • Problem: One root of 3x2−x+k=03x^2 - x + k = 0 is 53\frac{5}{3}. Find the other root and kk.

    • Identify coefficients: a=3a = 3, b=−1b = -1, c=kc = k

    • Formula for sum of roots: Sum=−ba=−−13=13\text{Sum} = -\frac{b}{a} = -\frac{-1}{3} = \frac{1}{3}

    • Let the known root be r1=53r_1 = \frac{5}{3} and the unknown root be r2r_2:

      • r1+r2=13r_1 + r_2 = \frac{1}{3}

      • 53+r2=13\frac{5}{3} + r_2 = \frac{1}{3}

      • r2=13−53=−43r_2 = \frac{1}{3} - \frac{5}{3} = -\frac{4}{3}

    • Formula for product of roots: Product=ca=k3\text{Product} = \frac{c}{a} = \frac{k}{3}

      • r_1 \times r_2 = \right(\frac{5}{3}\right)\right(-\frac{4}{3}\right) = -\frac{20}{9}

      • k3=−209\frac{k}{3} = -\frac{20}{9}

      • k=−203k = -\frac{20}{3}

Complex Solutions to Quadratic Equations

  • Comparing Real and Imaginary Roots Algebraically and Graphically:

    • Equation 1: x2−1=0x^2 - 1 = 0

    • Algebraic Solution: x^2 = 1 \text{\implies} x = \text{\pm} 1

    • Zeros/Roots/Solutions: \text{\pm} 1 (2 real roots)

    • Graph Analysis: The parabola y=x2−1y = x^2 - 1 intersects the x-axis at x=1x = 1 and x=−1x = -1

    • Equation 2: x2+1=0x^2 + 1 = 0

    • Algebraic Solution: x^2 = -1 \text{\implies} x = \text{\pm} i

    • Zeros/Roots/Solutions: \text{\pm} i (2 imaginary roots, no real roots)

    • Graph Analysis: The parabola y=x2+1y = x^2 + 1 never intersects the x-axis

  • Solving Quadratic Equations with Complex Solutions:

    • All solutions must be simplified and left in standard a+bia + bi form.

    • Example 1: Solve 3x2+x=−73x^2 + x = -7 using the quadratic formula

    • Standard form: 3x2+x+7=03x^2 + x + 7 = 0

    • Coefficients: a=3a = 3, b=1b = 1, c=7c = 7

    • Quadratic Formula: x = \frac{-b \text{\pm} \text{\sqrt{b^2 - 4ac}}}{2a}

    • Substitution: x = \frac{-1 \text{\pm} \text{\sqrt{1^2 - 4(3)(7)}}}{2(3)}

    • Simplify discriminant: x = \frac{-1 \text{\pm} \text{\sqrt{1 - 84}}}{6} = \frac{-1 \text{\pm} \text{\sqrt{-83}}}{6}

    • Result in a+bia + bi form: x = -\frac{1}{6} \text{\pm} \frac{i\text{\sqrt{83}}}{6}

    • Example 2: Solve 2x2=4x−72x^2 = 4x - 7 using the quadratic formula

    • Standard form: 2x2−4x+7=02x^2 - 4x + 7 = 0

    • Coefficients: a=2a = 2, b=−4b = -4, c=7c = 7

    • Quadratic Formula: x = \frac{-(-4) \text{\pm} \text{\sqrt{(-4)^2 - 4(2)(7)}}}{2(2)}

    • Simplify discriminant: x = \frac{4 \text{\pm} \text{\sqrt{16 - 56}}}{4} = \frac{4 \text{\pm} \text{\sqrt{-40}}}{4}

    • Simplify radical: \text{\sqrt{-40}} = 2i\text{\sqrt{10}}

    • Result in a+bia + bi form: x = \frac{4 \text{\pm} 2i\text{\sqrt{10}}}{4} = 1 \text{\pm} \frac{i\text{\sqrt{10}}}{2}

    • Example 3: Solve x2+6x+25=0x^2 + 6x + 25 = 0 by completing the square

    • Rearrange terms: x2+6x=−25x^2 + 6x = -25

    • Complete the square step: \frac{1}{2}(6) = 3 \text{\implies} 3^2 = 9

    • Add 99 to both sides: x2+6x+9=−25+9x^2 + 6x + 9 = -25 + 9

    • Perfect square binomial: (x+3)2=−16(x + 3)^2 = -16

    • Take square root of both sides: \text{\sqrt{(x + 3)^2}} = \text{\sqrt{-16}}

    • Solve for xx: x + 3 = \text{\pm} 4i \text{\implies} x = -3 \text{\pm} 4i

    • Example 4: Solve x2−2x+3=0x^2 - 2x + 3 = 0 by completing the square

    • Rearrange terms: x2−2x=−3x^2 - 2x = -3

    • Complete the square step: \frac{1}{2}(-2) = -1 \text{\implies} (-1)^2 = 1

    • Add 11 to both sides: x2−2x+1=−3+1x^2 - 2x + 1 = -3 + 1

    • Perfect square binomial: (x−1)2=−2(x - 1)^2 = -2

    • Take square root of both sides: \text{\sqrt{(x - 1)^2}} = \text{\sqrt{-2}}

    • Solve for xx: x - 1 = \text{\pm} i\text{\sqrt{2}} \text{\implies} x = 1 \text{\pm} i\text{\sqrt{2}}

Algebraic Operations with Complex Numbers

  • Multiplicative Inverse and Reciprocals:

    • Definition: To find the multiplicative inverse (reciprocal) of a complex expression, set it as the denominator under 11 and rationalize the denominator so that no imaginary terms remain at the bottom.

    • Example 1: Find the reciprocal of 3i3i

    • Reciprocal expression: 13i\frac{1}{3i}

    • Multiply numerator and denominator by ii: 13i×ii=i3i2\frac{1}{3i} \times \frac{i}{i} = \frac{i}{3i^2}

    • Substitute i2=−1i^2 = -1: i3(−1)=−i3\frac{i}{3(-1)} = -\frac{i}{3}

    • Example 2: Find the reciprocal of 2−4i2 - 4i

    • Reciprocal expression: 12−4i\frac{1}{2 - 4i}

    • Multiply numerator and denominator by the complex conjugate (2+4i)(2 + 4i): 1(2+4i)(2−4i)(2+4i)\frac{1(2 + 4i)}{(2 - 4i)(2 + 4i)}

    • Expand denominator: (2−4i)(2+4i)=4−16i2=4−16(−1)=4+16=20(2 - 4i)(2 + 4i) = 4 - 16i^2 = 4 - 16(-1) = 4 + 16 = 20

    • Simplify expression: 2+4i20=220+420i=110+15i\frac{2 + 4i}{20} = \frac{2}{20} + \frac{4}{20}i = \frac{1}{10} + \frac{1}{5}i

  • Algebraic Division of Complex Numbers:

    • Example 1: Divide 71−5i\frac{7}{1 - 5i}

    • Multiply numerator and denominator by conjugate 1+5i1 + 5i: 7(1+5i)(1−5i)(1+5i)\frac{7(1 + 5i)}{(1 - 5i)(1 + 5i)}

    • Expand numerator: 7+35i7 + 35i

    • Expand denominator: 1−25i2=1−25(−1)=261 - 25i^2 = 1 - 25(-1) = 26

    • Final result in standard form: 7+35i26=726+3526i\frac{7 + 35i}{26} = \frac{7}{26} + \frac{35}{26}i

    • Example 2: Divide (3 + 8i) \text{\div} (2 - 7i)

    • Fractional form: 3+8i2−7i\frac{3 + 8i}{2 - 7i}

    • Multiply numerator and denominator by conjugate 2+7i2 + 7i: (3+8i)(2+7i)(2−7i)(2+7i)\frac{(3 + 8i)(2 + 7i)}{(2 - 7i)(2 + 7i)}

    • Expand numerator: 3(2)+3(7i)+8i(2)+8i(7i)=6+21i+16i+56i2=6+37i−56=−50+37i3(2) + 3(7i) + 8i(2) + 8i(7i) = 6 + 21i + 16i + 56i^2 = 6 + 37i - 56 = -50 + 37i

    • Expand denominator: 22−(7i)2=4−49i2=4−49(−1)=532^2 - (7i)^2 = 4 - 49i^2 = 4 - 49(-1) = 53

    • Final result in standard form: −50+37i53=−5053+3753i\frac{-50 + 37i}{53} = -\frac{50}{53} + \frac{37}{53}i