Chapter 4.6, Problem 47HP

To determine

To find:

The value of $m$ by using the given quadratic equation.

Answer to Problem 47HP

  $m=-0.75$

Given:

The quadratic equation

  $x^2+x+m+1=0$

Concept used:

The number is the square root of the quadratic formula.

  $b^2-4ac$

The discriminant formula

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

The quadratic equation

  $x^2+x+m+1=\mathrm{0.......................}\left(1\right)$

Since the equation has one solution, its discriminant value is $0$ .

Comparing the quadratic equation $a{x}^2+bx+c=0$

  $a=1,b=1,c=m+1$

Putting the value of $a,b,c$ in equation (2)

  $\begin{array}{l}0=b^2-4ac\\ 0=1^2-4\left(1\right)\left(m+1\right)\\ 0=1-4m-4\\ 3=-4m\\ m=-\frac{3}{4}\\ m=-0.75\end{array}$

Therefore,

The value of quadratic function is $m=-0.75$ .

Explanation of Solution

Given:

The quadratic equation

  $x^2+x+m+1=0$

Concept used:

The number is the square root of the quadratic formula.

  $b^2-4ac$

The discriminant formula

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

The quadratic equation

  $x^2+x+m+1=\mathrm{0.......................}\left(1\right)$

Since the equation has one solution, its discriminant value is $0$ .

Comparing the quadratic equation $a{x}^2+bx+c=0$

  $a=1,b=1,c=m+1$

Putting the value of $a,b,c$ in equation (2)

  $\begin{array}{l}0=b^2-4ac\\ 0=1^2-4\left(1\right)\left(m+1\right)\\ 0=1-4m-4\\ 3=-4m\\ m=-\frac{3}{4}\\ m=-0.75\end{array}$

Therefore,

The value of quadratic function is $m=-0.75$ .