Solve the Quadratic Inequality: x²-3x+4<0 Step by Step

The problem requires us to solve the inequality $x^2 - 3x + 4 < 0$.

To solve the inequality, we first consider the corresponding quadratic equation $x^2 - 3x + 4 = 0$ and find its roots.

Calculate the discriminant $\Delta$:
$\Delta = b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7$.

The discriminant $\Delta = -7$ is less than zero, indicating that the quadratic equation has no real roots. This implies that the quadratic expression $x^2 - 3x + 4$ does not change sign and is either always positive or always negative.

Next, evaluate the sign of $x^2 - 3x + 4$. For $x = 0$, the expression is $0^2 - 3 \cdot 0 + 4 = 4$, which is positive. Therefore, the expression is always positive for all real $x$.

Since $x^2 - 3x + 4$ is always positive, there is no $x$ for which $x^2 - 3x + 4 < 0$ holds true.

Therefore, the solution to the inequality is that there is no solution, which corresponds to option 4: "There is no solution."