Convert to Interval Notation 3x^2-2x+1<0

Problem

3*x2−2*x+1<0

Solution

1. Identify the quadratic inequality and its coefficients. For the quadratic a*x2+b*x+c we have a=3 b=−2 and c=1

2. Calculate the discriminant to determine the nature of the roots using the formula D=b2−4*a*c

D=(−2)2−4*(3)*(1)

D=4−12

D=−8

3. Analyze the discriminant and the leading coefficient. Since D<0 the quadratic 3*x2−2*x+1 has no real roots and never crosses the x-axis.

4. Determine the sign of the quadratic. Since the leading coefficient a=3 is positive, the parabola opens upward and lies entirely above the x-axis.

3*x2−2*x+1>0

5. Evaluate the inequality 3*x2−2*x+1<0 Since the expression is always positive for all real numbers, there are no values of x that satisfy the inequality.

Final Answer

3*x2−2*x+1<0=∅

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