Question

Decide how many solutions the equation has. $$x^{2}-2 x+4=0$$

Short answer

The equation \(x^{2}-2 x+4=0\) has no real solutions.

Step-by-step solution

Identify coefficients a, b and c

In the quadratic equation \(ax^{2} + bx + c = 0\), the coefficient \(a\) is the number in front of \(x^{2}\), \(b\) in front of \(x\) and \(c\) is the constant. For the equation \(x^{2} - 2x + 4 = 0\), the coefficients are: \(a = 1\), \(b = -2\), and \(c = 4\).

Calculate the Discriminant

Using the formula for discriminant \(D = b^{2}-4ac\) with \(a = 1\), \(b = -2\), and \(c = 4\), we substitute those values in and get \(D = (-2)^{2}-4*1*4\).

Determine the Number of Solutions

From the previous step, we calculate \(D = 4 - 16 = -12\). Since \(D < 0\), there are no real solutions for the given equation.