Question

Find the number of real solutions of the equation$4x^2+5x+\frac{13}{8}=0$ by using discriminant.

Short answer

The equation has no real solutions.

Step-by-step solution

Step 1. From discriminant method of quadratic equation.

An equation of the form$a{x}^2+bx+c=0$ where$a\ne 0$ has discriminant$\left(D\right)=b^2-4ac$ in which three particular cases arises.

1. If $D>0$, the equation has two distinct real solutions.
2. If $D<0$, the equation has no real solution.
3. If $D=0$, the equation has exactly one real solution.

Step 2. Evaluate discriminant of given quadratic.

The discriminant of given quadratic equation $4x^2+5x+\frac{13}{8}=0$is evaluated as,

$$\begin{gathered} D={\left(5\right)}^2-4\left(4\right)\left(\frac{13}{8}\right) \\ =25-26 \\ =-1 \\ <0....\left(1\right) \end{gathered}$$

Step 3. Description of steps.

Since$D<0$ from (1), therefore the equation has no real solutions.