T1W6L3

#methods#lesson#ae#quadratics

the quadratic formula §

• there are methods for determining solutions to non-factorable quadratics, you can do it by using the quadratic formula. $x=-b\pm \frac{\sqrt{b^2-4ac}}{2a}$
• quadratic formula is derived from solving the general formula for completing the square.
• y intercept at (0, c): (0, -3)
• x intercept at y =0:
  • $2x^2+4x-3=0$
  • $x=-4\pm \frac{\sqrt{4^2-4(2)(-3)}}{2(2)}$
  • $=-4\pm \frac{\sqrt{16+24}}{4}$
  • $=-4\pm \frac{\sqrt{40}}{4}$
  • $=-\frac{2\pm \sqrt{10}}{2}$
  • $(\frac{-2-\sqrt{10}}{2},0)$, $(0,\frac{-2\pm \sqrt{10}}{2})$

discriminant §

• when solving quadratic equations, you can find either two, one or no real solutions.
• graphically these corresponds to the x-intercepts, where the function cross the x-axis (where y=0)
  • the number of solutions can be quickly identified by referring back to the quadratic formula.
  • if $b^2-4ac>0$, the square root is non-zero (two real solutions)
  • if $b^2-4ac=0$, the square root is zero (one real solution)
  • if $b^2-4ac<0$, the square root cannot be taken (no real solution)
  • $\Delta$ is the discriminant.
• consider the equation $x^2+18x+k+7=0$
  • a) find the values of k if the equation has no solution.
    • $18^2-4(1)(k+7)<0$
    • $324-4k-28<0$
    • $296-4k<0$
    • $296<4k$
    • $k>74$
    • b) k = 75
  • consider the equation $2x^2-2x=x-1$
    • $2x^2-3x+1=0$
    • a) find the discriminant
      • $-3^2-4(2)(1)=9-8=1$
    • b) $\Delta >0⟹$ 2 SOLS
  • the discriminant can also be used to check the nature of solutions. for a, b and c are rational numbers:
    • if $\Delta$ is a perfect square, then there are 2 rational solutions.
    • if $\Delta =0$, then there is one rational solution.
    • if $\Delta$ is not a perfect square and $\Delta >0$, then there are 2 irrational solutions.
    • consider the equation $x^2+22x+120=0$
      • a) find the value of the discriminant
        • $22^2-4(1)(120)=484-480=4$
      • b) using your answer from part a, determine whether the solutions to the equation are rational or irrational.
        • irrational.