After being rearranged and simplified, which two of the following equations
could be solved using the quadratic formula?
A. 5x + 2x - 4 = 2x2
B. x2 - 6x - 7 = 2x
C. 5x2 – 3x+ 10 = 2x2
D. 2x2 – 3x+ 10 = 2x2 + 21

A. 5x + 2x - 4 = 2x2  <- Yes

x = 7/4 + sqrt(17)/4 or x = 7/4 - sqrt(17)/4

B. x^2 - 6x - 7 = 2x    <- Yes

x = 4 + sqrt(23) or x = 4 - sqrt(23)

C: 5x^2 – 3x+ 10 = 2x2    <- Yes

x = 1/2 + 1/2 i sqrt(37/3) or x = 1/2 - 1/2 i sqrt(37/3)  

D . 2x^2 – 3x+ 10 = 2x^2 + 21x = -11/3     <- No

Step-by-step explanation:

Solve for x over the real numbers:

5 x + 2 x - 4 = 2 x^2

5 x + 2 x - 4 = 7 x - 4:

7 x - 4 = 2 x^2

Subtract 2 x^2 from both sides:

-2 x^2 + 7 x - 4 = 0

Divide both sides by -2:

x^2 - (7 x)/2 + 2 = 0

Subtract 2 from both sides:

x^2 - (7 x)/2 = -2

Add 49/16 to both sides:

x^2 - (7 x)/2 + 49/16 = 17/16

Write the left hand side as a square:

(x - 7/4)^2 = 17/16

Take the square root of both sides:

x - 7/4 = sqrt(17)/4 or x - 7/4 = -sqrt(17)/4

Add 7/4 to both sides:

x = 7/4 + sqrt(17)/4 or x - 7/4 = -sqrt(17)/4

Add 7/4 to both sides:

x = 7/4 + sqrt(17)/4 or x = 7/4 - sqrt(17)/4

Solve for x over the real numbers:

x^2 - 6 x - 7 = 2 x

Subtract 2 x from both sides:

x^2 - 8 x - 7 = 0

Add 7 to both sides:

x^2 - 8 x = 7

Add 16 to both sides:

x^2 - 8 x + 16 = 23

Write the left hand side as a square:

(x - 4)^2 = 23

Take the square root of both sides:

x - 4 = sqrt(23) or x - 4 = -sqrt(23)

Add 4 to both sides:

x = 4 + sqrt(23) or x - 4 = -sqrt(23)

Add 4 to both sides:

x = 4 + sqrt(23) or x = 4 - sqrt(23)

Solve for x:

5 x^2 - 3 x + 10 = 2 x^2

Subtract 2 x^2 from both sides:

3 x^2 - 3 x + 10 = 0

Divide both sides by 3:

x^2 - x + 10/3 = 0

Subtract 10/3 from both sides:

x^2 - x = -10/3

Add 1/4 to both sides:

x^2 - x + 1/4 = -37/12

Write the left hand side as a square:

(x - 1/2)^2 = -37/12

Take the square root of both sides:

x - 1/2 = 1/2 i sqrt(37/3) or x - 1/2 = -1/2 i sqrt(37/3)

Add 1/2 to both sides:

x = 1/2 + 1/2 i sqrt(37/3) or x - 1/2 = -1/2 i sqrt(37/3)

Add 1/2 to both sides:

x = 1/2 + 1/2 i sqrt(37/3) or x = 1/2 - 1/2 i sqrt(37/3)

Solve for x over the real numbers:

2 x^2 - 3 x + 10 = 2 x^2 + 21

Subtract 2 x^2 + 21 from both sides:

-3 x - 11 = 0

Multiply both sides by -1:

3 x + 11 = 0

Subtract 11 from both sides:

3 x = -11

Divide both sides by 3:

x = -11/3

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