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.
[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.
[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.
[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.
[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.
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.
