Question

use the discriminant to find the nature of the solutions to the following quadratic equation3x^2 -2x-4=0Two different irrational number solutionsOne repeated irrational number solutiontwo different rational number solutionsOne repeated rational number solutiontwo imaginary number solution

Answer 1

The quadratic equation is

3x^2 - 2x - 4

using completing the square method

firstly, to solve a quadratic equation the coefficient x^2 must be 1

divide through by 3

3x^2/3 -2x/3 - 4/3 = 0

make the linear expression a perfect square

x^2 - 2x/3 = 4/3

make -2x/3 a perfect square

x^2 - 2x/3 x 1/ 2 = 4/3

x^2 - 2x/6 = 4/3

square -2x/6

x^2 -(2x/6)^2 = 4/3

add the perfect square to both sides

x^2 - (2x/6)^2 = 4/3 + (2/6)^2

(x - 2/6)^2 = 4/3 + 4/36

solve the left hand side of the equation

4/3 + 4/36

lcm is 36

4 * 12 + 4 *1 / 36

48 + 4 / 36

52/36

(x - 2/6)^2 = 52/36

find the squareroot of both sides

`\begin{gathered} \text{x -2/6 }\pm\text{ }\sqrt[]{(52)/(36)_{}} \\ x\text{ = 2/6 }\pm\text{ }\frac{\sqrt[]{52}}{6} \\ \end{gathered}`

square root of 52 is 7.211

x = 2 + 7.211 / 6 0r 2 - 7.211 / 6

x = 9.211 / 6 or -5.211/6

x = 1.53 or -0.8685

the answer is two different rational number solutions