Answer:
C) 81; 2; ³/₂; -3
Discriminant = 81
Number of solutions = 2
x = ³/₂
x = -3
Step-by-step explanation:
The discriminant is defined as the expression b² - 4ac, which appears under the square root sign in the quadratic formula.
The value of the discriminant determines the nature of the solutions to the quadratic equation:
[tex]\boxed{\begin{array}{l}\underline{\sf Discriminant}\\\\b^2-4ac\\\\\textsf{when $ax^2+bx+c=0$}\\\\\textsf{$b^2-4ac > 0 \implies$ two real solutions}\\\textsf{$b^2-4ac=0 \implies$ one real solution}\\\textsf{$b^2-4ac < 0 \implies$ no real solutions}\end{array}}[/tex]
Given quadratic equation:
[tex]2x^2+3x-9=0[/tex]
Therefore, the coefficients are:
Substitute the values of a, b and c into the discriminant formula:
[tex]b^2-4ac=3^2-4(2)(-9)\\\\b^2-4ac=9-(8)(-9)\\\\b^2-4ac=9+72\\\\b^2-4ac=81[/tex]
Therefore, the discriminant is 81.
As the discriminant of the given equation is greater than zero, there are 2 real solutions.
To solve 2x² + 3x - 9 = 0, use the quadratic formula:
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\\\x=\dfrac{-3\pm\sqrt{81}}{4}\\\\\\x=\dfrac{-3\pm9}{4}[/tex]
Therefore, the two solutions are:
[tex]\textsf{Solution 1:}\quad x=\dfrac{-3+9}{4}=\dfrac{6}{4}=\dfrac{3}{2}\\\\\\\textsf{Solution 2:}\quad x=\dfrac{-3-9}{4}=\dfrac{-12}{4}=-3[/tex]
