Question
Evaluate the discriminant and determine the number and type of solutions to the eq \[ 4 x^{2}-5 x+3=0 \] Part: \( 0 / 2 \) Part 1 of 2
Ask by Luna Davies. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The discriminant is –23, so the equation has two complex solutions.
Solution
The quadratic equation is 4x² – 5x + 3 = 0.
Step 1: Identify the coefficients:
a = 4, b = –5, c = 3.
Step 2: Calculate the discriminant using Δ = b² – 4ac:
Δ = (–5)² – 4(4)(3) = 25 – 48 = –23.
Since the discriminant is negative (Δ = –23), the equation has two complex (non-real) solutions.
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Simplify this solution Extra Insights
To evaluate the discriminant of the equation \( 4x^{2}-5x+3=0 \), we use the formula for the discriminant, \( D = b^2 - 4ac \), where \( a = 4 \), \( b = -5 \), and \( c = 3 \). Calculating it: \[ D = (-5)^2 - 4 \times 4 \times 3 = 25 - 48 = -23 \] Since the discriminant \( D = -23 \) is less than zero, this means that the quadratic equation has two complex solutions (no real solutions). You can think of these solutions as "imaginary friends" that exist in a different realm!
