Answer :
To determine the nature of the roots of the quadratic equation [tex]\( x^2 - 8x + 14 = 0 \)[/tex], we use the discriminant. The discriminant [tex]\( D \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by the formula:
[tex]\[ D = b^2 - 4ac \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are coefficients of the quadratic equation.
For the given equation [tex]\( x^2 - 8x + 14 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = 14 \)[/tex]
Substituting these values into the formula for the discriminant, we get:
[tex]\[ D = (-8)^2 - 4 \cdot 1 \cdot 14 \][/tex]
[tex]\[ D = 64 - 56 \][/tex]
[tex]\[ D = 8 \][/tex]
Now, we will use the value of the discriminant to determine the nature of the roots:
1. If [tex]\( D > 0 \)[/tex], there are two distinct real solutions.
2. If [tex]\( D = 0 \)[/tex], there is one repeated real solution.
3. If [tex]\( D < 0 \)[/tex], there are two complex solutions that are not real.
Since [tex]\( D = 8 \)[/tex], which is greater than 0, there are two distinct real solutions. Further, we need to determine if these solutions are rational or irrational.
For the solutions to be rational, the discriminant itself must be a perfect square. However, 8 is not a perfect square (the square root of 8 is approximately 2.828, which is not an integer). Therefore, the solutions are not rational.
Thus, the quadratic equation has two distinct irrational real solutions.
The correct answer is:
D. two irrational solutions
[tex]\[ D = b^2 - 4ac \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are coefficients of the quadratic equation.
For the given equation [tex]\( x^2 - 8x + 14 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = 14 \)[/tex]
Substituting these values into the formula for the discriminant, we get:
[tex]\[ D = (-8)^2 - 4 \cdot 1 \cdot 14 \][/tex]
[tex]\[ D = 64 - 56 \][/tex]
[tex]\[ D = 8 \][/tex]
Now, we will use the value of the discriminant to determine the nature of the roots:
1. If [tex]\( D > 0 \)[/tex], there are two distinct real solutions.
2. If [tex]\( D = 0 \)[/tex], there is one repeated real solution.
3. If [tex]\( D < 0 \)[/tex], there are two complex solutions that are not real.
Since [tex]\( D = 8 \)[/tex], which is greater than 0, there are two distinct real solutions. Further, we need to determine if these solutions are rational or irrational.
For the solutions to be rational, the discriminant itself must be a perfect square. However, 8 is not a perfect square (the square root of 8 is approximately 2.828, which is not an integer). Therefore, the solutions are not rational.
Thus, the quadratic equation has two distinct irrational real solutions.
The correct answer is:
D. two irrational solutions
Answer:
Step by-step explanation:
D=[tex]b^{2}-4ac[/tex]
for the given equation [tex]x^{2} -8x+14=0[/tex] we have
a=1
b=-18
c=14
so D=[tex](-8)^{2}-4(1)(14)=64-56=8[/tex]
the value of D is 8
if D>0 and is a perfect square there are two rational solutions.
if D>0 and is not a perfect square there are two irrational solutions.
if D=0 there is one repeated real root.
if D<0 there are two complex solutions.
Therefore the correct answer is option D.
since D=8.
