use the value of the discriminant to determine the number and type of roots for each equation to answer the following questions
2x^2-7x+9=0
2x^2-7x+9=0
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GPT-4o mini
To determine the number and type of roots for the quadratic equation \( 2x^2 - 7x + 9 = 0 \), we first need to identify the coefficients \( a \), \( b \), and \( c \).
In this equation:
- \( a = 2 \)
- \( b = -7 \)
- \( c = 9 \)
Next, we'll calculate the discriminant \( D \) using the formula:
\[
D = b^2 - 4ac
\]
Substituting the values of \( a \), \( b \), and \( c \):
\[
D = (-7)^2 - 4(2)(9)
\]
\[
D = 49 - 72
\]
\[
D = -23
\]
The discriminant \( D = -23 \) is less than 0.
According to the properties of the discriminant:
- If \( D > 0 \), there are two distinct real roots.
- If \( D = 0 \), there is one real root (a repeated root).
- If \( D < 0 \), there are two complex (non-real) roots.
Since \( D = -23 \) is less than 0, we conclude that the equation \( 2x^2 - 7x + 9 = 0 \) has **two complex (non-real) roots**.
In this equation:
- \( a = 2 \)
- \( b = -7 \)
- \( c = 9 \)
Next, we'll calculate the discriminant \( D \) using the formula:
\[
D = b^2 - 4ac
\]
Substituting the values of \( a \), \( b \), and \( c \):
\[
D = (-7)^2 - 4(2)(9)
\]
\[
D = 49 - 72
\]
\[
D = -23
\]
The discriminant \( D = -23 \) is less than 0.
According to the properties of the discriminant:
- If \( D > 0 \), there are two distinct real roots.
- If \( D = 0 \), there is one real root (a repeated root).
- If \( D < 0 \), there are two complex (non-real) roots.
Since \( D = -23 \) is less than 0, we conclude that the equation \( 2x^2 - 7x + 9 = 0 \) has **two complex (non-real) roots**.
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