Mathematico | Mathematico

By drawing a suitable sketch, explain why the equation

4 x^{2} - 4 x + 5 = 0

has no real solutions.

Hint

Sketch the parabola

y = 4 x^{2} - 4 x + 5

noting the position of its vertex.

Hint

The coefficient of $x^{2}$ is positive, so we know it is a convex parabola.

Solution

In complete square form, the equation of the parabola is

\begin{aligned} y & = 4 [x^{2} - x] + 5 \\ = 4 [{(x - \frac{1}{2})}^{2} - \frac{1}{4}] + 5 \\ = 4 {(x - \frac{1}{2})}^{2} - 1 + 5 \\ = 4 {(x - \frac{1}{2})}^{2} + 4 \end{aligned}

The vertex occurs at $(\frac{1}{2}, 4)$ . In addition, the parabola is convex (as the coefficient of $x^{2}$ is positive) and the $y$ -intercept is $5$ .

This enables us to make an accurate sketch

Parabola

From the sketch, it is clear that no point on the parabola has a $y$ coordinate of $0$ , and so $4 x^{2} - 4 x + 5 = 0$ can have no solutions.