Question

Write the discriminant of the equation $2x^2-5x-4=0$ and determine the nature of the roots.

Answer 1

Step 1: Comparing the given equation with the standard form of the quadratic equation

The given equation is $2x^2-5x-4=0$.

The standard form of the quadratic equation is $a{x}^2+bx+c=0$.

From the above two equations, the values are

$$\begin{gathered} a=2 \\ b=-5 \\ c=-4 \end{gathered}$$

Step 2: Finding the discriminant,$D$:

The formula for finding the discriminant is $D=b^2-4ac$.

Substituting the values,

$$\begin{gathered} \Rightarrow D=b^2-4ac \\ \Rightarrow D={(-5)}^2-4\times 2\times (-4) \\ \Rightarrow D=25+32 \\ \therefore D=57 \end{gathered}$$

Step 3: Nature of the root of the quadratic equation

The value of the discriminant decides the nature of the roots of the quadratic equation.

If $D>0,$ then the quadratic equation has real and unequal roots.

If $D=0,$ then the quadratic equation has real and equal roots.

If $D<0,$ then the quadratic equation has no real roots.

Since, the discriminant $D=57$ is greater than zero, the given quadratic equation $2x^2-5x-4=0$ has real and unequal roots.

Hence, the quadratic equation has real and unequal roots.