Chapter 1.9, Problem 4E

a.

To determine

To calculate: The equation $y=5x-x^2$ and $y=4$ ,gives the solution of the equation $5x-x^2=4$ .

  

Answer to Problem 4E

The solution of the equation $y=5x-x^2$ and $y=4$ is at which the curve touches the $x-axis$ is 1 and 4.

Explanation of Solution

Given information:

The equation $y=5x-x^2$ and $y=4$ , gives the solution of the equation $5x-x^2=4$ with the help of given graph.

Formula used:

To the help of middle term factorisation method.

Steps to use this method to solve a quadratic polynomial $a{x}^2-bx-c=0$ .

Step 1. Multiply the coefficient of $x$ and the constant $c$ .

Step 2. Now to get the coefficient of $x$ , use the resultant value of multiplication of $x$ and $c$ by multiplication or subtraction or division .

Step 3. Now we have 4 terms now, take common from 1^st two terms and take common from another two common and from there two factors were come.

Step 3. Take either one bracket is equal to zero or another is equal to zero.

Step 4. Now the value of $x$ comes.

Calculation:

The graph of the equation $y=5x-x^2$ and $y=4$ is sketched:

So find the roots of the equation $y=5x-x^2$ and $y=4$ ,

Rewrite the equation: $5x-x^2-4=0$

Rearrange the equation in a polynomial form and take ‘-’ common from the equation:

  $\begin{array}{l}5x-x^2-4=0\\ x^2-5x+4=0\end{array}$

Further simplify the equation:

  $\begin{array}{c}{x}^2-5x+4=0\\ x^2-4x-x+4=0\end{array}$

Now to take common:

  $\begin{array}{c}{x}^2-4x-x+4=0\\ x\left(x-4\right)-1\left(x-4\right)=0\end{array}$

Either $x-1=0$ or $x-4=0$ :

Simplify further as: $x=1$ and $x=4$

The solution of the equation $y=5x-x^2$ and $y=4$ is at which the curve touches the $x-axis$ is 1 and 4.

b.

To determine

To calculate: The solution of the equations $5x-x^2\succ 4$ .

Answer to Problem 4E

The solution of the inequality $5x-x^2\succ 4$ is $x\in \left(-\infty ,1\right)\cup \left(4,-\infty \right)$ .

Explanation of Solution

Given information:

The inequality $5x-x^2\succ 4$ .

Formula used:

To the help of middle term factorisation method.

Steps to use this method to solve a quadratic polynomial $a{x}^2-bx-c=0$ .

Step 1. Multiply the coefficient of $x$ and the constant $c$ .

Step 2. Now to get the coefficient of $x$ , use the resultant value of multiplication of $x$ and $c$ by multiplication or subtraction or division .

Step 3. Now we have 4 terms now, take common from 1^st two terms and take common from another two common and from there two factors were come.

Step 3. Take either one bracket is equal to zero or another is equal to zero.

Step 4. Now the value of $x$ comes.

Calculation:

The inequality $5x-x^2\succ 4$ .

So find the roots of the inequality $5x-x^2\succ 4$ ,

Rewrite the inequality: $5x-x^2-4=0$

Rearrange the equation in a polynomial form and take ‘-’ common from the equation:

  $\begin{array}{l}5x-x^2-4=0\\ x^2-5x+4=0\end{array}$

Further simplify the equation:

  $\begin{array}{c}{x}^2-5x+4=0\\ x^2-4x-x+4=0\end{array}$

Now to take common:

  $\begin{array}{c}{x}^2-4x-x+4=0\\ x\left(x-4\right)-1\left(x-4\right)=0\end{array}$

Either $x-1=0$ or $x-4=0$ :

Simplify further as: $x=1$ and $x=4$

Thus, the solution of the inequality $5x-x^2\succ 4$ is the area of curve above the line $y=4$

  $x\in \left(-\infty ,1\right)\cup \left(4,-\infty \right)$ or $\left[1,4\right]$ .