Question
Use the discriminant to tell whether the solutions of this equation are real or complex. \[ 5 x^{2}-7 x+1=0 \]
Ask by Salazar Stephens. in the United States
Mar 17,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The discriminant is 29, which is greater than 0, so the equation has two distinct real solutions.
Solution
1. Write down the quadratic equation and identify coefficients:
\[
5x^2 - 7x + 1 = 0
\]
Here, \( a = 5 \), \( b = -7 \), and \( c = 1 \).
2. Write the formula for the discriminant:
\[
\Delta = b^2 - 4ac
\]
3. Substitute the values of \( a \), \( b \), and \( c \) into the formula:
\[
\Delta = (-7)^2 - 4(5)(1)
\]
4. Simplify the expression:
\[
\Delta = 49 - 20 = 29
\]
5. Analyze the discriminant:
Since \(\Delta = 29 > 0\), the equation has two distinct real solutions.
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Simplify this solution Extra Insights
To determine whether the solutions to the equation \(5x^2 - 7x + 1 = 0\) are real or complex, we can use the discriminant formula, which is given by \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\). For this equation, we have: - \(a = 5\) - \(b = -7\) - \(c = 1\) Now, let's calculate the discriminant: \[ D = (-7)^2 - 4 \cdot 5 \cdot 1 = 49 - 20 = 29 \] Since the discriminant \(D\) is positive (\(D = 29 > 0\)), this means that there are two distinct real solutions to the equation. So, the solutions are indeed real and not complex!
