Question

Write the discriminant of the equation $3x^2-17x+20=0$ 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 $3x^2-17x+20=0$.

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

From the above two equations, the values are

$$\begin{gathered} a=3 \\ b=-17 \\ c=20 \end{gathered}$$

Step 2: Finding the discriminant $\left(D\right)$

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

Substituting values,

$$\begin{gathered} \Rightarrow D={(-17)}^2-4\times 3\times 20 \\ \Rightarrow D={(-17)}^2-240 \\ \Rightarrow D=289-240 \\ \therefore D=49 \end{gathered}$$

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 $D=49$ is greater than zero, the given quadratic equation $3x^2-17x+20=0$ has real and unequal roots.

Hence, the quadratic equation has real and unequal roots.