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# Completing The Square

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What times what= 2 but when added= -4 When the linear-quadratic equation is -1/2x^2-2x+7=-2x+5.

Apr 8, 2020

#1
+20981
+1

Is this two different questions?

1)  What times what  =  2  but when added  =  -4:

What times what  =  2   --->   x · y  =  2

When added  =  -4        --->   x + y  =  -4

Solving  x · y  =  2  for  y:  y  =  2/x

Sustituting this into the other equation:  x + y  =  -4   --->   x + 2/x  =  -4

multiplying both sides of this equation by x:             x2 + 2  =  -4x

adding 4x to both sides:                                    x2 + 4x + 2  =  0

Using the quadratic formula when  a = 1, b = 4, and c = 2:   x  =  [ -4 +/- sqrt( 42 - 4·1·2 ) ] / ( 2·1)

x  =  [ -4 +/- sqrt(8) ] / 2

x  =  [ -4 +/- 2sqrt(2) ] / 2

One answer:  x = -2 + sqrt(2)

Other answer:  x = -2 - sqrt(2)

2)  When the linear-quadratic equation is -1/2x2 - 2x + 7  =  -2x + 5

-1/2x2 + 2  =  0

multiply both sides by -2:                                  x2 - 4  =  0

(x + 2)(x - 2)  =  0

Apr 9, 2020

#1
+20981
+1

Is this two different questions?

1)  What times what  =  2  but when added  =  -4:

What times what  =  2   --->   x · y  =  2

When added  =  -4        --->   x + y  =  -4

Solving  x · y  =  2  for  y:  y  =  2/x

Sustituting this into the other equation:  x + y  =  -4   --->   x + 2/x  =  -4

multiplying both sides of this equation by x:             x2 + 2  =  -4x

adding 4x to both sides:                                    x2 + 4x + 2  =  0

Using the quadratic formula when  a = 1, b = 4, and c = 2:   x  =  [ -4 +/- sqrt( 42 - 4·1·2 ) ] / ( 2·1)

x  =  [ -4 +/- sqrt(8) ] / 2

x  =  [ -4 +/- 2sqrt(2) ] / 2

One answer:  x = -2 + sqrt(2)

Other answer:  x = -2 - sqrt(2)

2)  When the linear-quadratic equation is -1/2x2 - 2x + 7  =  -2x + 5

-1/2x2 + 2  =  0

multiply both sides by -2:                                  x2 - 4  =  0

(x + 2)(x - 2)  =  0