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  1. 1Solving 7(x−2)=3x+10
  2. 2Solving a rational equation
  3. 3Solving 5−3x≤17
  4. 5Slope between two points
  5. 6Line through two points
  6. 7Slope and intercept from 3x−4y=12
  7. 8Parallel line through a point
  8. 10Solving a system by substitution
  9. 11Elimination, 2 equations
  10. 14Exponent rules, same base
  11. 15Fractional exponent, 27^(2/3)
  12. 17Simplifying root sums
  13. 20Solving 3^(x+1)=81
  14. 21Factoring x²−5x−24
  15. 23Difference of squares
  16. 24Quadratic formula
  17. 26Vertex of a parabola
  18. 27Discriminant of a quadratic
  19. 28One solution, discriminant=0
  20. 30Expanding a binomial times trinomial
  21. 31Evaluating f(-2)
  22. 34Exponential growth, 5% a year
  23. 35Half-life decay
  24. 37Solving |2x−5|=9
  25. 39Simplifying a rational expression

27Discriminant of a quadratic

Find the discriminant of 3x2−4x+5=0\displaystyle 3x^2 - 4x + 5 = 0. How many real solutions does the equation have?

The discriminant is Δ = (−4)² − 4(3)(5) = 16 − 60 = −44. Since Δ is negative, the equation has no real solutions (the parabola never touches the x-axis).

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