Chapter 9.6, Problem 45PPE

a)

To determine

To evaluate: the discriminant and the solution of equation.

Answer to Problem 45PPE

The discriminant is 16 and the solution is $5,1$ .

Explanation of Solution

Given: Equation is given as $x^2-6x+5=0$ .

Taking the given equation,

  $x^2-6x+5=0$ .

Finding the discriminant,

  $\begin{array}{c}d=b^2-4ac\\ ={\left(-6\right)}^2-4\left(1\right)\left(5\right)\\ =36-20\\ =16\end{array}$

So, the discriminant is 16.

Now finding the solution,

  $\begin{array}{c}x=\frac{6\pm \sqrt{36-20}}{2}\\ x=\frac{6\pm 4}{2}\\ x=5,1\end{array}$

Hence, the solution is $5,1$ .

b)

To determine

To evaluate: the discriminant and the solution of equation.

Answer to Problem 45PPE

The discriminant is 81 and the solution is $5,-4$ .

Explanation of Solution

Given: Equation is given as $x^2-x-20=0$ .

Taking the given equation,

  $x^2-x-20=0$ .

Finding the discriminant,

  $\begin{array}{c}d=b^2-4ac\\ ={\left(-1\right)}^2-4\left(1\right)\left(-20\right)\\ =1+80\\ =81\end{array}$

So, the discriminant is 81.

Now finding the solution,

  $\begin{array}{c}x=\frac{1\pm \sqrt{1+80}}{2}\\ x=\frac{1\pm 9}{2}\\ x=5,-4\end{array}$

Hence, the solution is $5,-4$ .

c)

To determine

To evaluate: the discriminant and the solution of equation.

Answer to Problem 45PPE

The discriminant is 73 and the solution is $\frac{7\pm \sqrt{73}}{2}$ .

Explanation of Solution

Taking the given equation,

  $2x^2-7x-3=0$ .

Finding the discriminant,

  $\begin{array}{c}d=b^2-4ac\\ ={\left(-7\right)}^2-4\left(2\right)\left(-3\right)\\ =49+24\\ =73\end{array}$

So, the discriminant is 73.

Now finding the solution,

  $\begin{array}{c}x=\frac{7\pm \sqrt{49+24}}{2}\\ x=\frac{7\pm \sqrt{73}}{2}\end{array}$

Hence, the solution is $\frac{7\pm \sqrt{73}}{2}$ .

c)

To determine

To explain: if the discriminant is a perfect square then solutions are rational or irrational.

Answer to Problem 45PPE

Equation will have rational roots.

Explanation of Solution

It can be seen from the part b), that when the discriminant is perfect square then the equation has rational roots.

So, equation will have rational roots.