Question

При каких значениях t уравнение 2x^2-7x+t-8=0 имеет два корня?

Answer 1

Квадратное уравнение имеет 2 действительных корня, если дискриминант больше 0.

D=49-4×2×(t-8)=49-8t+64=113-8t

113-8t>0
-8t>-113
t<113/8
t<14,125

Answer 2

Solving the Quadratic Equation 2x^2 - 7x + t - 8 = 0

To find the values of t for which the equation 2x^2 - 7x + t - 8 = 0 has two roots, we can use the discriminant formula.

The discriminant (denoted as Δ) of a quadratic equation ax^2 + bx + c = 0 is given by the formula:

Δ = b^2 - 4ac

Where: - a = 2 - b = -7 - c = t - 8

For the equation to have two distinct real roots, the discriminant Δ must be greater than 0.

Calculating the Discriminant

Using the discriminant formula, we can calculate the discriminant Δ for the given quadratic equation:

Δ = (-7)^2 - 4 * 2 * (t - 8)

Finding the Values of t

For the equation to have two distinct real roots, the discriminant Δ must be greater than 0. Therefore, we need to solve the inequality:

Δ > 0

Substituting the expression for Δ, we get:

(-7)^2 - 4 * 2 * (t - 8) > 0

Solving this inequality will give us the range of values for t for which the equation 2x^2 - 7x + t - 8 = 0 has two distinct real roots.

Conclusion

By solving the inequality derived from the discriminant, we can find the values of t for which the given quadratic equation has two distinct real roots.