2x²-3x+4>0;
To solve the inequality 2x² - 3x + 4 > 0, you can follow these steps:
- Start by finding the roots of the quadratic equation 2x² - 3x + 4 = 0. To find the roots, you can use the quadratic formula, which states that the roots of a quadratic equation ax² + bx + c = 0 are given by:
x = (-b ± √(b² - 4ac)) / (2a)
Applying this formula to our equation of 2x² - 3x + 4 = 0, we have:
x = (-(-3) ± √((-3)² - 4 * 2 * 4)) / (2 * 2) x = (3 ± √(9 - 32)) / 4 x = (3 ± √(-23)) / 4
Since the discriminant (the value inside the square root) is negative, there are no real roots for this quadratic equation. This means that the quadratic equation does not intersect the x-axis and the parabola opens upwards.
- Since the quadratic equation does not intersect the x-axis, it means that the value of 2x² - 3x + 4 is always positive or always negative. We need to determine when it is positive.
To find this out, you can check the sign of the coefficient of x², which is positive, i.e., a = 2. Since a > 0, the parabola opens upwards and the sign of the quadratic equation is positive when x is between the roots of the equation, which is always.
This means that 2x² - 3x + 4 > 0 for all values of x.
So, the solution to the inequality 2x² - 3x + 4 > 0 is the set of all real numbers or the interval (-∞, ∞).
