1

Problem

Solve the quadratic equation \$2x^2 + 3x - 4 = 0\$ by completing the square.

2

Step

Start with the given quadratic equation

\$2x^2 + 3x - 4 = 0\$

3

Step

Move the constant term (-4) to the right side of the equation:

\$2x^2 + 3x = 4\$

4

Step

Divide the entire equation by the coefficient of \$x^2\$ to make the leading coefficient 1 . Then after simplification:

\$(2x^2)/2 + (3/2)x = 4/2\$

\$x^2+(3/2)x = 2\$

5

Step

To complete the square, take half of the coefficient of x, square it, and add it to both sides of the equation, then after simplification:

\$x^2 + (3/2)x + (3/4)^2 = 2 + (3/4)^2\$

\$x^2 + (3/2)x + 9/(16) = (32)/(16) + 9/(16)\$

6

Step

Combining the fractions:

\$x^2 + (3/2)x + 9/(16) = (41)/(16)\$

\$(x + 3/4)^2 = (41)/(16)\$

7

Step

Take the square root of both sides of the equation, considering both positive and negative square roots. Then after simplification:

\$x + 3/4 = ± \sqrt (41/16)\$

\$x = -3/4 ± (\sqrt 41)/4\$

8

Step

When we take the positive and negative square roots:

\$x = (-3 +(\sqrt 41))/4\$ and \$x = (-3 - (\sqrt 41))/4\$

9

Solution

Therefore, the solutions to the quadratic equation \$2x^2 + 3x - 4 = 0\$, obtained by completing the square are \$x = (-3 + \sqrt (41)) / 4\$ and \$x = (-3 - \sqrt (41)) / 4\$.

10

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

11

Choice-A

The solutions given for the quadratic equations are not applicable

Wrong \$x = (-4 + \sqrt (41)) / 3\$ and \$x = (-4 - \sqrt (41)) / 3\$

12

Choice-B

The solutions given for the quadratic equations are not applicable

Wrong \$x = (-3 - \sqrt (41)) / 4\$ and \$x = (3 + \sqrt (41)) / 4\$

13

Choice-C

The solutions given for the quadratic equations are not applicable

Wrong \$x = (4 + \sqrt (41)) / 3\$ and \$x = (4 - \sqrt ( 41)) / 3\$

14

Choice-D

The pair of solutions align with the provided quadratic equations

Correct \$x = (-3 + \sqrt (41)) / 4\$ and \$x = (-3 - \sqrt (41)) / 4\$

15

Answer

Option

D

16

Sumup

Please summarize choices