Solve -7x^2 + 13x - 31 = 0: animated lesson

1. 1. The problem
   Solving an equation

   $-7~{ x}^{2}+13~ x-31 = 0$
2. 2. Steps, part 1
   1. $$-7~{ x}^{2}+13~ x-31 = 0$$
      Start with the equation
   2. $$-7x^{2} + 13x - 31 = 0$$
      Move every term to one side to get standard form
   3. $$\Delta = b^2 - 4ac = 13^2 - 4 \cdot (-7) \cdot (-31) = -699$$
      Compute the discriminant
   4. $$$$
      Δ < 0, so there are no real roots; the two roots are complex conjugates
3. 3. Steps, part 2
   1. $$x = \frac{-13 \pm \sqrt{-699}}{-14} = \frac{-13 \pm \sqrt{699}\,i}{-14}$$
      Apply the quadratic formula with √(−1) = i
   2. $$x = \frac{13 \pm \sqrt{699}\,i}{14}$$
      Simplify
4. 4. The answer
   $$x_1 = \frac{13 - \sqrt{699}\,i}{14},\quad x_2 = \frac{13 + \sqrt{699}\,i}{14}$$

   Δ = -699; x = (13 - √699·i)/14 or x = (13 + √699·i)/14

5. 5. Check
   • ✓Vieta: sum of roots = 1.8571429 = −b/a = 1.8571429; product = 4.4285714 = c/a = 4.4285714

Solve quadratics by factoring, completing the square or the quadratic formula, including complex roots.

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