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solve each equation by using the quadratic formula. 19. $9x^{2}+7x + 3=…

Question

solve each equation by using the quadratic formula. 19. $9x^{2}+7x + 3=0$ 20. $6n^{2}-10n - 6=0$ 21. $6=-11a^{2}-5a$ 22. $-4r=-3 - 9r^{2}$ 23. $3p^{2}+8=0$ 24. $4r=3 + 9r^{2}$ 25. $-4t^{2}+t=2$ 26. $-2x^{2}=3x - 1$ for each equation, find the value of the discriminant.

Explanation:

Response
  1. For equation \(9x^{2}+7x + 3=0\) (problem 19):
  • Explanation:
  • Step 1: Identify the coefficients in the quadratic - equation \(ax^{2}+bx + c = 0\)
  • For the equation \(9x^{2}+7x + 3=0\), we have \(a = 9\), \(b = 7\), and \(c = 3\).
  • Step 2: Calculate the discriminant \(\Delta=b^{2}-4ac\)
  • Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \(\Delta=(7)^{2}-4\times9\times3=49 - 108=-59\).
  • Step 3: Use the quadratic formula \(x=\frac{-b\pm\sqrt{\Delta}}{2a}\)
  • Since \(\Delta=-59\lt0\), the solutions are complex numbers. \(x=\frac{-7\pm\sqrt{-59}}{2\times9}=\frac{-7\pm i\sqrt{59}}{18}\).
  1. For equation \(6n^{2}-10n - 6 = 0\) (problem 20):
  • Explanation:
  • Step 1: Identify the coefficients
  • First, divide the entire equation by \(2\) to simplify: \(3n^{2}-5n - 3 = 0\). Here, \(a = 3\), \(b=-5\), and \(c=-3\).
  • Step 2: Calculate the discriminant \(\Delta=b^{2}-4ac\)
  • \(\Delta=(-5)^{2}-4\times3\times(-3)=25 + 36 = 61\).
  • Step 3: Use the quadratic formula \(n=\frac{-b\pm\sqrt{\Delta}}{2a}\)
  • \(n=\frac{5\pm\sqrt{61}}{2\times3}=\frac{5\pm\sqrt{61}}{6}\).
  1. For equation \(6=-11a^{2}-5a\) (problem 21):
  • Explanation:
  • Step 1: Rewrite the equation in standard form \(ax^{2}+bx + c = 0\)
  • \(11a^{2}+5a + 6 = 0\), where \(a = 11\), \(b = 5\), and \(c = 6\).
  • Step 2: Calculate the discriminant \(\Delta=b^{2}-4ac\)
  • \(\Delta=(5)^{2}-4\times11\times6=25-264=-239\).
  • Step 3: Use the quadratic formula \(a=\frac{-b\pm\sqrt{\Delta}}{2a}\)
  • \(a=\frac{-5\pm\sqrt{-239}}{2\times11}=\frac{-5\pm i\sqrt{239}}{22}\).
  1. For equation \(-4r=-3 - 9r^{2}\) (problem 22):
  • Explanation:
  • Step 1: Rewrite the equation in standard form
  • \(9r^{2}-4r + 3 = 0\), where \(a = 9\), \(b=-4\), and \(c = 3\).
  • Step 2: Calculate the discriminant \(\Delta=b^{2}-4ac\)
  • \(\Delta=(-4)^{2}-4\times9\times3=16 - 108=-92\).
  • Step 3: Use the quadratic formula \(r=\frac{-b\pm\sqrt{\Delta}}{2a}\)
  • \(r=\frac{4\pm\sqrt{-92}}{2\times9}=\frac{4\pm2i\sqrt{23}}{18}=\frac{2\pm i\sqrt{23}}{9}\).
  1. For equation \(3p^{2}+8 = 0\) (problem 23):
  • Explanation:
  • Step 1: Identify the coefficients
  • Here, \(a = 3\), \(b = 0\), and \(c = 8\).
  • Step 2: Calculate the discriminant \(\Delta=b^{2}-4ac\)
  • \(\Delta=(0)^{2}-4\times3\times8=0 - 96=-96\).
  • Step 3: Use the quadratic formula \(p=\frac{-b\pm\sqrt{\Delta}}{2a}\)
  • \(p=\frac{0\pm\sqrt{-96}}{2\times3}=\frac{\pm4i\sqrt{6}}{6}=\frac{\pm2i\sqrt{6}}{3}\).
  1. For equation \(4r = 3+9r^{2}\) (problem 24):
  • Explanation:
  • Step 1: Rewrite the equation in standard form
  • \(9r^{2}-4r + 3 = 0\) (same as problem 22). \(a = 9\), \(b=-4\), \(c = 3\).
  • Step 2: Calculate the discriminant \(\Delta=b^{2}-4ac\)
  • \(\Delta=(-4)^{2}-4\times9\times3=16 - 108=-92\).
  • Step 3: Use the quadratic formula \(r=\frac{-b\pm\sqrt{\Delta}}{2a}\)
  • \(r=\frac{4\pm\sqrt{-92}}{2\times9}=\frac{4\pm2i\sqrt{23}}{18}=\frac{2\pm i\sqrt{23}}{9}\).
  1. For equation \(-4t^{2}+t = 2\) (problem 25):
  • Explanation:
  • Step 1: Rewrite the equation in standard form
  • \(4t^{2}-t + 2 = 0\), where \(a = 4\), \(b=-1\), and \(c = 2\).
  • Step 2: Calculate the discriminant \(\Delta=b^{2}-4ac\)
  • \(\Delta=(-1)^{2}-4\times4\times2=1 - 32=-31\).
  • **Step 3: Use the…

Answer:

  1. \(x=\frac{-7\pm i\sqrt{59}}{18}\)
  2. \(n=\frac{5\pm\sqrt{61}}{6}\)
  3. \(a=\frac{-5\pm i\sqrt{239}}{22}\)
  4. \(r=\frac{2\pm i\sqrt{23}}{9}\)
  5. \(p=\frac{\pm2i\sqrt{6}}{3}\)
  6. \(r=\frac{2\pm i\sqrt{23}}{9}\)
  7. \(t=\frac{1\pm i\sqrt{31}}{8}\)
  8. \(x=\frac{-3\pm\sqrt{17}}{4}\)