Chapter 7.3, Problem 32WE

To determine

The value of $\left(k\right)$ for which the equation $3x^2-6x-k=0$ has a real double root.

Answer to Problem 32WE

The value of $\left(k\right)$ is equal to $-3$ for a real double root condition.

Explanation of Solution

Given information: The given quadratic equation is,

  $3x^2-6x-k=\mathrm{0....}\left(\text{1}\right)$

Formula used: The discriminant $\left(D=b^2-4ac\right)$ of a quadratic equation $a{x}^2+bx+c=0$ where $\left(a\ne 0\right)$ should be equal to zero for the real double root condition.

Mathematically,

  $b^2-4ac=\mathrm{0....}\left(2\right)$

Calculation: From (1), evaluate discriminant $\left(D\right)$ and equate that to zero according to formula (2).

  $\begin{array}{c}D={\left(-6\right)}^2-4\left(3\right)\left(-k\right)\mathrm{....}\left\{\begin{array}{l}a=3,b=-6\\ \text{and }c=-k\end{array}\right\}\\ 0=36+12k\mathrm{....}\left(D=0\text{ for double root}\right)\\ \text{or, }12k=-36\\ k=-3\end{array}$

To determine

The value of $\left(k\right)$ for which the equation $3x^2-6x-k=0$ has two different real roots.

Answer to Problem 32WE

The value of $\left(k\right)$ is greater than $-3$ for two different real roots

Explanation of Solution

Given information: The given quadratic equation is $3x^2-6x-k=0$ .

Formula used: The discriminant $\left(D=b^2-4ac\right)$ of a quadratic equation $a{x}^2+bx+c=0$ where $\left(a\ne 0\right)$ should be greater than zero for two different root equation.

Mathematically,

  $b^2-4ac>\mathrm{0....}\left(3\right)$

Calculation: From (1), evaluate discriminant $\left(D\right)$ and apply the formula (3).

  $\begin{array}{c}{\left(-6\right)}^2-4\left(3\right)\left(-k\right)>\mathrm{0....}\left\{\begin{array}{l}a=3,b=-6\\ \text{and }c=-k\end{array}\right\}\\ 36+12k>\mathrm{0....}\left(D>0\text{from }\left(3\right)\right)\\ \text{or, }12k>-36\\ k>-3\end{array}$

To determine

The value of $\left(k\right)$ for which the equation $3x^2-6x-k=0$ has the imaginary roots.

Answer to Problem 32WE

The value of $\left(k\right)$ is less than $-3$ for imaginary roots.

Explanation of Solution

Given information: The given quadratic equation is $3x^2-6x-k=0$ .

Formula used: The discriminant $\left(D=b^2-4ac\right)$ of a quadratic equation $a{x}^2+bx+c=0$ where $\left(a\ne 0\right)$ should be less than zero for imaginary roots.

Mathematically,

  $b^2-4ac<\mathrm{0....}\left(4\right)$

Calculation: From (1), evaluate discriminant $\left(D\right)$ and apply formula (4).

  $\begin{array}{c}{\left(-6\right)}^2-4\left(3\right)\left(-k\right)<\mathrm{0....}\left\{\begin{array}{l}a=3,b=-6\\ \text{and }c=-k\end{array}\right\}\\ 36+12k<\mathrm{0....}\left(D<0\text{from }\left(4\right)\right)\\ \text{or, }12k<-36\\ k<-3\end{array}$