Answer:
1. Real, Rational, and Equal
2. Real, Rational, and Distinct
3. Real, Rational, and Distinct
4. Imaginary
5. Real, Irrational, and Distinct
6. Real, Rational, and Distinct
7. Imaginary
8. Imaginary
9. Real, Irrational, and Distinct
10. Real, Rational, and Distinct
Step-by-step explanation:
The formula for discriminant is , we use this to determine the number of roots of solutions in a quadratic equation. There are three cases:
1.) a = 1, b = 6, c = 9
Discriminant =
=
= 81 – 80
= 1
Nature: Real, Rational, and Distinct. The equation has 2 real solutions.
3.) a = 2, b = -10, c = 8
= 100 – 64
= 36
4.) a = 1, b = 5, c = 10
= 25 – 40
= -15
Nature: Imaginary. The equation has 0 real solutions.
5.) a = 1, b = 6, c = 3
= 36 – 12
= 24
Nature: Real, Irrational, and Distinct. The equation has 2 real solutions.
6.) a = 2, b = 6, c = 4
= 36 – 32
= 4
7.) a = 3, b = -5, c = 4
= 25 – 48
= -23
8.) a = 9, b = -6, c = 9
= 36 – 324
= -288
9.) a = 10, b = -4, c = -8
= 16 + 320
= 336
10.) a = 3, b = -2, c = -5
= 4 + 60
= 64
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Answers & Comments
Verified answer
Answer:
1. Real, Rational, and Equal
2. Real, Rational, and Distinct
3. Real, Rational, and Distinct
4. Imaginary
5. Real, Irrational, and Distinct
6. Real, Rational, and Distinct
7. Imaginary
8. Imaginary
9. Real, Irrational, and Distinct
10. Real, Rational, and Distinct
Step-by-step explanation:
DISCRIMINANT: NATURE OF THE ROOTS
The formula for discriminant is
, we use this to determine the number of roots of solutions in a quadratic equation. There are three cases:
1.) a = 1, b = 6, c = 9
Discriminant =![b^{2} -4ac b^{2} -4ac](https://tex.z-dn.net/?f=b%5E%7B2%7D%20-4ac)
=![6^{2} -4(1)(9)\\ ></p><p> = 36 – 36</p><p> <strong>= 0</strong></p><p><u><em>Nature:</em></u> <strong>Real, Rational, and Equal</strong>. <em>The equation has 1 real solution.</em></p><p></p><p><strong>2.) a = 1, b = 9, c = 20</strong></p><p><u><em>Discriminant</em></u><em> =</em> <img src=](https://tex.z-dn.net/?f=6%5E%7B2%7D%20-4%281%29%289%29%5C%5C)
=![9^{2} -4(1)(20) 9^{2} -4(1)(20)](https://tex.z-dn.net/?f=9%5E%7B2%7D%20-4%281%29%2820%29)
= 81 – 80
= 1
Nature: Real, Rational, and Distinct. The equation has 2 real solutions.
3.) a = 2, b = -10, c = 8
Discriminant =![b^{2} -4ac b^{2} -4ac](https://tex.z-dn.net/?f=b%5E%7B2%7D%20-4ac)
=![(-10)^{2} -4(2)(8) (-10)^{2} -4(2)(8)](https://tex.z-dn.net/?f=%28-10%29%5E%7B2%7D%20-4%282%29%288%29)
= 100 – 64
= 36
Nature: Real, Rational, and Distinct. The equation has 2 real solutions.
4.) a = 1, b = 5, c = 10
Discriminant =![b^{2} -4ac b^{2} -4ac](https://tex.z-dn.net/?f=b%5E%7B2%7D%20-4ac)
=![5^{2} -4(1)(10) 5^{2} -4(1)(10)](https://tex.z-dn.net/?f=5%5E%7B2%7D%20-4%281%29%2810%29)
= 25 – 40
= -15
Nature: Imaginary. The equation has 0 real solutions.
5.) a = 1, b = 6, c = 3
Discriminant =![b^{2} -4ac b^{2} -4ac](https://tex.z-dn.net/?f=b%5E%7B2%7D%20-4ac)
=![6^{2} -4(1)(3) 6^{2} -4(1)(3)](https://tex.z-dn.net/?f=6%5E%7B2%7D%20-4%281%29%283%29)
= 36 – 12
= 24
Nature: Real, Irrational, and Distinct. The equation has 2 real solutions.
6.) a = 2, b = 6, c = 4
Discriminant =![b^{2} -4ac b^{2} -4ac](https://tex.z-dn.net/?f=b%5E%7B2%7D%20-4ac)
=![6^{2} -4(2)(4) 6^{2} -4(2)(4)](https://tex.z-dn.net/?f=6%5E%7B2%7D%20-4%282%29%284%29)
= 36 – 32
= 4
Nature: Real, Rational, and Distinct. The equation has 2 real solutions.
7.) a = 3, b = -5, c = 4
Discriminant =![b^{2} -4ac b^{2} -4ac](https://tex.z-dn.net/?f=b%5E%7B2%7D%20-4ac)
=![(-5)^{2} -4(3)(4) (-5)^{2} -4(3)(4)](https://tex.z-dn.net/?f=%28-5%29%5E%7B2%7D%20-4%283%29%284%29)
= 25 – 48
= -23
Nature: Imaginary. The equation has 0 real solutions.
8.) a = 9, b = -6, c = 9
Discriminant =![b^{2} -4ac b^{2} -4ac](https://tex.z-dn.net/?f=b%5E%7B2%7D%20-4ac)
=![(-6)^{2} -4(9)(9) (-6)^{2} -4(9)(9)](https://tex.z-dn.net/?f=%28-6%29%5E%7B2%7D%20-4%289%29%289%29)
= 36 – 324
= -288
Nature: Imaginary. The equation has 0 real solutions.
9.) a = 10, b = -4, c = -8
Discriminant =![b^{2} -4ac b^{2} -4ac](https://tex.z-dn.net/?f=b%5E%7B2%7D%20-4ac)
=![(-4)^{2} -4(10)(-8) (-4)^{2} -4(10)(-8)](https://tex.z-dn.net/?f=%28-4%29%5E%7B2%7D%20-4%2810%29%28-8%29)
= 16 + 320
= 336
Nature: Real, Irrational, and Distinct. The equation has 2 real solutions.
10.) a = 3, b = -2, c = -5
Discriminant =![b^{2} -4ac b^{2} -4ac](https://tex.z-dn.net/?f=b%5E%7B2%7D%20-4ac)
=![(-2)^{2} -4(3)(-5) (-2)^{2} -4(3)(-5)](https://tex.z-dn.net/?f=%28-2%29%5E%7B2%7D%20-4%283%29%28-5%29)
= 4 + 60
= 64
Nature: Real, Rational, and Distinct. The equation has 2 real solutions.
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