Chapter 9.5, Problem 36PPS

To determine

The value of the discriminant of the equation $2x^2-5x+20=0$ and find the number of real solutions.

Answer to Problem 36PPS

  $D=-135,\text{ no real roots}$

Explanation of Solution

Given information:

The given equation, $2x^2-5x+20=0$

Concept used:

• The discriminant is given by, $D=b^2-4ac$
• For a quadratic equation $a{x}^2+bx+c=0$ the number of roots depend on:
• If discriminant is negative, $b^2-4ac<0$, then there is no real roots
• If discriminant is zero, $b^2-4ac=0$, then there are two same real roots
• If discriminant is positive, $b^2-4ac>0$, then there are two distinct real roots

Calculation:

To find the discriminant of the given quadratic equation first rewrite the given equation in standard form So it gives

  $2x^2-5x+20=0$

Now compare this equation with the general equation, $a{x}^2+bx+c=0$ to get the value of the variables, $a=2,b=-5,c=20$ . So, the discriminant of the given equation is

  $\begin{array}{c}D=b^2-4ac\\ ={\left(-5\right)}^2-4\left(2\right)\left(20\right)\\ =25-160\\ =-135\end{array}$

So, the discriminant of the given equation is $-135$ .

Since the discriminant is negative and less zero, so there are no real and distinct roots of the given equation.