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Quadratic equations with complex roots worksheets Grades 10–11

Solve quadratics whose roots are complex numbers with the quadratic formula.

The front of the free quadratic equations with complex roots worksheet
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Free · Worked example · 9 problems · Answer key · US Letter

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How to do it

Use the quadratic formula. The square root of a negative number gives i. Write x = a ± bi.

x2+10x+41=0x=−5±4ix^{2} + 10x + 41 = 0\quad x = -5 \pm 4i
a =1,b=10,c=41\text{a }= 1,\quad b = 10,\quad c = 41
102−4(1)(41)=−6410^{2} - 4(1)(41) = -64b² − 4ac is negative
x=−10±−642x = \dfrac{-10 \pm \sqrt{-64}}{2}x = (−b ± √(b² − 4ac)) ÷ 2a
x=−10±8i2x = \dfrac{-10 \pm 8i}{2}√−64 = 8i
x=−5±4ix = -5 \pm 4idivide both parts

Example problems

  1. 2x2−8x+16=02x^{2} - 8x + 16 = 0
  2. 2x2−20x+100=02x^{2} - 20x + 100 = 0
  3. x2−6x+34=0x^{2} - 6x + 34 = 0
  4. 2x2+24x+144=02x^{2} + 24x + 144 = 0

Quadratic and polynomial functions

  1. Vertex form of a quadratic
  2. Finding the vertex from standard form
  3. The discriminant
  4. Quadratic equations with complex roots
  5. Polynomial long division
  6. Synthetic division
  7. The remainder theorem
  8. The factor theorem
  9. The rational root theorem