To determine the number and type of solutions of the quadratic equation [tex]\( -7x^2 - 5x + 5 = 0 \)[/tex], we need to 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]
For the given quadratic equation, the coefficients are:
- [tex]\( a = -7 \)[/tex]
- [tex]\( b = -5 \)[/tex]
- [tex]\( c = 5 \)[/tex]
Substituting these values into the discriminant formula:
[tex]\[
D = (-5)^2 - 4(-7)(5)
\][/tex]
Calculate the first part of the discriminant:
[tex]\[
(-5)^2 = 25
\][/tex]
Now, calculate the second part:
[tex]\[
4(-7)(5) = 4 \cdot (-7) \cdot 5 = -140
\][/tex]
Next, combining both parts to find the discriminant:
[tex]\[
D = 25 - (-140) = 25 + 140 = 165
\][/tex]
The discriminant [tex]\( D = 165 \)[/tex].
Now, let's analyze the discriminant to determine the number and type of solutions:
- If [tex]\( D > 0 \)[/tex] and [tex]\( D \)[/tex] is a perfect square, the quadratic equation has two distinct rational solutions.
- If [tex]\( D > 0 \)[/tex] and [tex]\( D \)[/tex] is not a perfect square, the quadratic equation has two distinct irrational solutions.
- If [tex]\( D = 0 \)[/tex], the quadratic equation has one real solution.
- If [tex]\( D < 0 \)[/tex], the quadratic equation has two complex solutions.
Since [tex]\( D = 165 \)[/tex] and 165 is greater than 0 but not a perfect square, the quadratic equation has two distinct irrational solutions.
Therefore, the correct answer is:
There are two distinct irrational solutions.
