Chapter 1.8, Problem 62E

Solution

To write: The quadratic equation in standard form that has discriminant $-10$ .

The required quadratic equation in standard form is $7.5x^2+\sqrt{5}x+0.5=0$ where $a\ne 0$ .

Given:

The discriminant $\left(D\right)$ of the quadratic equation is $-10$ .

Concept used:

Any quadratic equation of the form $a{x}^2+bx+c=0\text{ where }a\ne 0$ has the discriminant $D=b^2-4ac$ .

Calculation:

Consider the quadratic equation in standard form is $a{x}^2+bx+c=0\text{ where }a\ne 0$ . Since, the discriminant of it is $-10$ . Therefore,

  $b^2-4ac=-10$ .

Now, consider the parameters $a,b\text{and }c$ such that $b^2-4ac=-10$ . Therefore, by hit and trial method ${\left(\sqrt{5}\right)}^2-4\left(7.5\right)\left(0.5\right)=-10$ .

Comparing ${\left(\sqrt{5}\right)}^2-4\left(7.5\right)\left(0.5\right)=-10$ with $b^2-4ac=-10$ gives, $b=\sqrt{5},a=7.5\text{ and }c=0.5$ .

Hence, the required quadratic equation is written as $7.5x^2+\sqrt{5}x+0.5=0$ where $a\ne 0$ .