Chapter 9.5, Problem 57HP

To determine

A positive discrimenant one with a negative discrimnant and one with a zero discriminant

Answer to Problem 57HP

  $D=0\text{ there are two same real roots}$

  $D=18.25\text{there are two distinct real roots}$

  $D=-17.94\text{there is no real roots}$

Explanation of Solution

Given information:

A quadratic function

Concept used:

• The discriminant is given by, $D=b^2-4ac$
• For a quadratic equation $a{x}^2+bx+c=0$ the number of roots depend on:
• If discriminant is negative, $b^2-4ac<0$, then there is no real roots
• If discriminant is zero, $b^2-4ac=0$, then there are two same real roots
• If discriminant is positive, $b^2-4ac>0$, then there are two distinct real roots

Calculation:

To find the zero discriminant let us assume a equation $x^2-8x+16=0$

To find the discriminant of the given quadratic equation first rewrite the given equation in standard form So it gives

  $x^2-8x+16=0$

Now compare this equation with the general equation, $a{x}^2+bx+c=0$ to get the value of the variables, $a=1,b=-8,c=16$ . So, the discriminant of the given equation is

  $\begin{array}{c}D=b^2-4ac\\ ={\left(-8\right)}^2-4\left(1\right)\left(16\right)\\ =64-64\\ =0\end{array}$

So, the discriminant of the given equation is $0$ .

Since the discriminant is zero, so there are there are two same real roots of the given equation.

Let us assume a another equation $2x^2-\frac{5}{2}x-\frac{3}{2}=0$

To find the discriminant of the given quadratic equation first rewrite the given equation in standard form So it gives

  $2x^2-\frac{5}{2}x-\frac{3}{2}=0$

Now compare this equation with the general equation, $a{x}^2+bx+c=0$ to get the value of the variables, $a=2,b=\frac{-5}{2},c=\frac{-3}{2}$ . So, the discriminant of the given equation is

  $\begin{array}{c}D=b^2-4ac\\ ={\left(-\frac{5}{2}\right)}^2-4\left(2\right)\left(\frac{-3}{2}\right)\\ =\frac{25}{4}+12\\ =6.25+12\\ =18.25\end{array}$

So, the discriminant of the given equation is $D=18.25$ .

Since the discriminant is positive and greater than zero, so there are there are two distinct real roots of the given equation.

Let us assume a another equation $4.25x+3=-3x^2$

To find the discriminant of the given quadratic equation first rewrite the given equation in standard form by add both sides by $3x^2$ So it gives

  $3x^2+4.25x+3=0$

Now compare this equation with the general equation, $a{x}^2+bx+c=0$ to get the value of the variables, $a=3,b=4.25,c=3$ . So, the discriminant of the given equation is

  $\begin{array}{c}D=b^2-4ac\\ ={\left(4.25\right)}^2-4\left(3\right)\left(3\right)\\ =18.06-36\\ =-17.94\end{array}$

So, the discriminant of the given equation is $D=-17.94$ .

Since the discriminant is negative and less than zero, so there are there are no real roots of the given equation.