Find the value(s) of c if the given equation has no real roots. Explain.
2x^2 – 7x + c = 0
To determine the value(s) of c for which the equation 2x^2 - 7x + c = 0 has no real roots, we need to consider the discriminant of the quadratic equation.
The discriminant, denoted as ∆, is given by the formula: ∆ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0.
For the given equation, the coefficients are a = 2, b = -7, and c = c.
If the discriminant is negative (∆ < 0), then the quadratic equation will have no real roots. Thus, we need to find the c values that make the discriminant negative.
∆ = (-7)^2 - 4(2)(c)
∆ = 49 - 8c
For the quadratic equation to have no real roots, the discriminant must be less than zero:
∆ < 0
49 - 8c < 0
Solving this inequality:
8c > 49
c > 49/8
Therefore, the value(s) of c for which the given equation has no real roots is c > 49/8.