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State whether the roots of each equation are real and unequal, real, and equal or imaginary and
unequal. (10 points)
1. 5x²-3x + 4 = 0
2. 4x² +2(x+4) = 0
3.6x² + 11x + 7 = 0
4. 2x² + 5x + 4 = 0 5. 25x² = 4x - 8

Sagot :

Aeiveryidn

Answer:

[tex]\tt {1.\:}[/tex] Unequal and imaginary

[tex]\tt {2.\:}[/tex] Unequal and imaginary

[tex]\tt {3.\:}[/tex] Unequal and imaginary

[tex]\tt {4.\:}[/tex] Unequal and imaginary

[tex]\tt {5.\:}[/tex] Unequal and imaginary

Solution:

We need to solve the following using the discriminant: [tex]\tt\underline {b^{2} -4ac }[/tex]. In order to find the nature of their roots.

[tex]\tt {1.\:\:5^2-3x+4=0}[/tex]

  • [tex]\tt {b^{2} -4ac = (-3)^2 -4(5)(4)}[/tex]
  • [tex]\tt {b^{2} -4ac = 9 -80}[/tex]
  • [tex]\tt {b^{2} -4ac = -71}[/tex]

Since [tex]\tt {-71 < 0}[/tex], the nature of roots is Unequal and imaginary.

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[tex]\tt {2.\:\:\:4x^2+2x+8=0\:(4^2+2(x+4)=0)}[/tex]

  • [tex]\tt {b^{2} -4ac = 2^2 -4(4)(8)}[/tex]
  • [tex]\tt {b^{2} -4ac = 4 -128}[/tex]
  • [tex]\tt {b^{2} -4ac = -124}[/tex]

Since [tex]\tt {-124 < 0}[/tex], the nature of roots is Unequal and imaginary.

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[tex]\tt {3.\:\:\:6x^2+11x+7=0}[/tex]

  • [tex]\tt {b^{2} -4ac = 11^2 -4(6)(7)}[/tex]
  • [tex]\tt {b^{2} -4ac = 121 -168}[/tex]
  • [tex]\tt {b^{2} -4ac = -47}[/tex]

Since [tex]\tt {-47 < 0}[/tex], the nature of roots is Unequal and imaginary.

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[tex]\tt {4.\:\:\:2x^2+5x+4=0}[/tex]

  • [tex]\tt {b^{2} -4ac = 5^2 -4(2)(4)}[/tex]
  • [tex]\tt {b^{2} -4ac = 25 -32}[/tex]
  • [tex]\tt {b^{2} -4ac = -7}[/tex]

Since [tex]\tt {-7 < 0}[/tex], the nature of roots is Unequal and imaginary.

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[tex]\tt {5.\:\:\:25x^2-4x+8=0}[/tex]

  • [tex]\tt {b^{2} -4ac = (-4)^2 -4(25)(8)}[/tex]
  • [tex]\tt {b^{2} -4ac =16 -800}[/tex]
  • [tex]\tt {b^{2} -4ac = -784}[/tex]

Since [tex]\tt {-784 < 0}[/tex], the nature of roots is Unequal and imaginary.

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Note:

  • If the discriminant is negative. The equation is unequal and imaginary.
  • If the discriminant is equal to 0. The equation is Real and Equal
  • If the discriminant is greater than 0. the equation is Real and unequal.