Chapter EP, Problem 5.6.1EP

(a)

To determine

To Find: The value of the discriminant.

Answer to Problem 5.6.1EP

  $-3$

Explanation of Solution

Given:

  $x^2+7x+13=0$

Formula Used:

  $D=b^2-4ac$

Calculation:

  $\begin{array}{l}{x}^2+7x+13=0\\ a=1\\ b=7\\ c=13\\ D=49-4\times 1\times 13=49-52=-3\end{array}$

Conclusion:

Thus the discriminant is $-3$

(b)

To determine

To Find: The number and type of roots.

Answer to Problem 5.6.1EP

Imaginary Roots

Explanation of Solution

Given:

  $x^2+7x+13=0$

Formula Used:

  $\begin{array}{l}D=b^2-4ac\\ D>0\text{Two distinct real roots}\\ D=0\text{Both roots are real and equal}\\ D<0\text{Imaginary}\end{array}$

Calculation:

  $\begin{array}{l}{x}^2+7x+13=0\\ a=1\\ b=7\\ c=13\\ D=49-4\times 1\times 13=49-52=-3<0\end{array}$

Conclusion:

Thus the roots are imaginary.

(c)

To determine

To Find: The solution of the given quadratic equation using quadratic formula.

Answer to Problem 5.6.1EP

  $\frac{-7+\sqrt{3}i}{4}or\frac{-7+\sqrt{3}i}{4}$

Explanation of Solution

Given:

  $x^2+7x+13=0$

Formula Used:

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

  $\begin{array}{l}{x}^2+7x+13=0\\ a=1\\ b=7\\ c=13\\ D=49-4\times 1\times 13=49-52=-3<0\\ x=\frac{-7\pm \sqrt{D}}{4\left(1\right)}\\ \Rightarrow x=\frac{-7+\sqrt{3}i}{4}or\frac{-7+\sqrt{3}i}{4}\end{array}$

Conclusion:

Thus the solution of the given quadratic equation is $\frac{-7+\sqrt{3}i}{4}or\frac{-7+\sqrt{3}i}{4}$