Solve by completing the square: 2x^2 - 7x -15 = 0?

Question

Solve by completing the square:

2x^2 - 7x -15 = 0

Thanks!

Answer 1

2x² - 7x - 15 = 0

Isolate the x's on the left and the numbers on the right:

2x² - 7x = 15

Divide by the coefficient of x² (which is 2) so that there's a 1 in front of it:

x² - 7/2 x = 15/2

Now you take the coefficient of x (-7/2), divide by 2 (-7/4), and square it (49/16), and add to both sides:

x² - 7/2 x + 49/2 = 15/2 + 49/16 = 120/16 + 49/16 = 169/16

Now on the left you have a perfect square:

(x - 7/4)² = 169/16

Take square root of both sides, remembering to do plus and minus:

x - 7/4 = ±√(169/16) = ±13/4

And solve for x:

x = 7/4 ± 13/4 = (7 ± 13)/4

x = (7 + 13)/4 = 20/4 = 5

x = (7 - 13)/4 = -6/4 = -3/2

Answer 2

2x² - 7x - 15 = 0

*/ For reference only x² - 7/2 x - 15/2 = 0
The product of the roots should be -15/2 (ie: a * c) and the sum of the roots should be 7/2 (ie : -(b/a) ) /*

The roots of the quadratic equation are

x = (-b ± √b² - 4ac) /2a
where a , b and c are coefficients of each of the expression
ie: a = 2 b = -7 c = -15

x = ( - (-7) ± √ (-7)² - 4(2)(-15) ) / 2(2)
x = (7 ± √49 + 120) / 4
x =(7 ± √169) /4
x = (7 ± 13 )/4

x = (7+13)/4 = 20/4 = 5

x = (7-13)/4 = -6/4 = -3/2

*/ For reference only
Sum of the roots = 5 + (-3/2) = 5- 3/2 = 7/2 and Product of the roots = 5 * (-3/2) = -15/2 . Verified ... /*

Answer 3

Hey there!

Recall that in completing the square for x^2+bx=c, the quadratic will then be x^2+bx+(b/2)^2=c+(b/2)^2.

Note that x^2+bx+(b/2)^2 will factor into (x+b/2)^2.

Now here come's the fun part.

2x^2-7x-15=0 -->
x^2-(7/2)x-(15/2)=0 -->
x^2-(7/2)x=15/2 -->
x^2-(7/2)x+(7/4)^2=15/2+(7/4)^2 -->
x^2-(7/2)x+49/16=15/2+49/16 -->
(x-7/4)^2=15/2+49/16 -->
(x-7/4)^2=120/16+49/16 -->
(x-7/4)^2=169/16 -->
x-7/4=±sqrt(169/16) -->
x-7/4=±13/4 -->
x=7/4±13/4 -->
x=7/4+13/4 or 7/4-13/4 -->
x=20/4 or x=-6/4 -->
x=5 or x=-3/2

So the solutions are x=-3/2 and x=5.

Hope it helps!

Answer 4

It is easy next time solve it for yourself
2x^2 - 7x = 15
x^2 - 7/2 x = 15/2
use (1/2 b)^2
then add it to both sides
x^2 - 7/2 x + 49/16 = 120/16 + 49/16
(x - 7/4)^2 = 169/16
x - 7/4 =(+ or -) 13/4
x = 20/4 = 5
or x = - 6/4 = - 3/2 = - 1.5
try hard next time

Answer 5

2x^2 - 7x - 15 = 0
x^2 - 7x/2 - 15/2 = 0
x^2 - 7x/2 = 15/2
So: x^2 - 7x/2 + ___ = 15/2 + ___
~~
x^2 - 7x/2 + 49/16 = 120/16 + 49/16 = 169/16
(x - 7/4)^2 = 169/16
x - 7/4 = +- 13/4
x = (7 +- 13)/4
x = 5 and x = -3/2

Answer 6

2(x + (3/2))(x - 5)

Thus your roots are at -3/2 and 5

Answer 7

2x^2 - 7x -15 = 0
x^2 -3.5x = 7.5
x^2 -3.5x + 3.0625 = 7.5 +3.0625
(x-1.75)^2 = 10.5625
(x-1.75) = +/- sqrt(10.5625)
x = 1.75 +/- sqrt(10.5625)
x = 1.75 +/- 3.25
x = 5 and x = -1.5