Question

\(\sf \Large \hspace{50 pt} \text{Discriminant of Quadratic Function}\\ \sf \hspace{100 pt} \text{by Data_LG2}\)

In a quadratic function (\(\sf ax^2+bx+c=0\)), the number or types of roots it will have can be determined by its \(\sf discriminant\). This topic will be discussed below.

Source: My Peabrain.

Answer 1

\(\sf \Large \color{red}{Discriminant}\) came from the famous \(\sf \text{quadratic formula}: \\ \hspace{-75 pt} \Large x=\frac{{-b \pm \sqrt{\color{blue}{b^2-4ac}}}}{2a}\)


The discriminant is given as: \(\sf \large \color{red}{D}= \color{blue}{b^2-4ac}\)
**Notice that this formula can be seen in the radical from the quadratic formula.

This determines the types of roots that the function will have depending on the following conditions:

\(\hspace{20 pt} \begin{array}{|l|c|}
\hline
\sf \large \text{If D is...}&\sf \large \text{then quadratic function will have...}\\
\hline
\hspace{5 pt} \sf < 0 &\sf \text{no real roots}\\
\hline
\hspace{5 pt} \sf = 0 &\sf \text{one real root}\\
\hline
\hspace{5 pt} \sf > 0 &\sf \text{two real roots}\\
\hline
\end{array}\)

Answer 2

\(\sf \Large \text{Equation is Given}\)

1. Make sure that the equation is in the standard form \(\sf ax^2 + bx +c =0\), otherwise rearrange the equation.

2. Identify the values of a, b, and c from the equation.

3. Plug-in the coefficient values into the discriminant formula to solve for the discriminant. From here, you can determine the type of roots the quadratic function will have.

Answer 3

\(\sf \huge \color{blue}{\text{Sample Questions:}}\\
\sf \text{How many solutions will the following quadratic functions will have?} \)


A. \(\sf \color{#999900}{-3x^2 + 2x + 8 = 0}\)

1. The equation is in std form.
2. Values for a, b, and c are -3, 2, and 8 respectively.
3. Plug it into the formula: \(\sf b^2-4ac\\ \sf (2)^2-4(-3)(8) \\ \sf = 100\)

Therefore, since the value of the discriminant is 100, which is greater than zero and it is a perfect square of 10, the quadratic function will have \(\sf \text{Two Rational Solutions}\)



B. \(\sf \color{#999900}{2x^2 - 5x = -20} \)

1. The equation is not in std form, so let's rearrange it a little bit.
\(\hspace{15 pt} \sf 2x^2 - 5x \color{red}{+ 20} = -20 \color{red}{+ 20} \\ \hspace{15 pt} \sf 2x^2 -5x + 20 = 0\)

2. Values for a, b, and c are 2, -5, and 20 respectively.
3. Plug it into the formula: \(\sf b^2-4ac\\ \sf (-5)^2-4(2)(20) \\ \sf = -135\)
Since the discrimant is -135, which is less than zero, this quadratic function will have \(\sf \text{No Real Solutions}\) (imaginary).




C. \(\sf \color{#999900}{2x^2 - 4x + 2 = 0}\)

1. 1. The equation is in std form.
2. Values for a, b, and c are 1, -6, and 13 respectively.
3. Plug it into the formula: \(\sf b^2-4ac\\ \sf (4)^2-4(2)(2) \\ \sf = 0\)

The discriminant is equal to zero, this means that this quadratic function has \(\sf \text{One Real Solution}\).

Answer 4

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Answer 5

That's nice and clean, thank you for sharing with us (-:

Answer 6

Oh wow. Very nice explanation

Answer 7

veryyyyy nice!!!