Question 1123897
your equation is:


2x^2 + 13x = 15


divide both sides of this equation by 2 to get:


x^2 + 6.5x = 7.5


take half the coefficient of the x term to get:


(x + 3.25)^2 - 3.25^2 = 7.5


simplify to get:


(x + 3.25)^2 - 10.5625 = 7.5


add 10.5625 to both sides of the equation to get:


(x + 3.25)^2 = 7.5 + 10.5625


simplify to get:


(x + 3.25)^2 = 18.0625


take the square root of both sides of this equation to get:


x + 3.25 = 4.25


subtract 3.25 from both sides of this equation to get:


x = 1


that's your solution.


confirm by replacing x with 1 in the original equation.


2x^2 + 13x = 15 becomes 2*1^2 + 13*1 = 15


simplify to get 2 + 13 = 15


combine like terms to get 15 = 15


this confirms the solution is correct.


here's a reference on completing the squares method of solving quadratic equations.


<a href = "https://www.purplemath.com/modules/sqrquad.htm" target = "_blank">https://www.purplemath.com/modules/sqrquad.htm</a>


when completing the square, the coefficient of the x^2 term needs to be 1.


that's why i divided both sides of the equation by 2 when i started.


the equation then became x^2 + 6.5x = 7.5


i then took half the coefficient of the x term to get:


(x + 3.25)^2 - 3.25^2 = 7.5


the reason for this is as follows:


(x + 3.25)^2 = x^2 + 6.5x + 3.25^2


since i wanted (x + 3.5)^2 to be equal to x^2 + 6.5x, i needed to subtract 3.25^2 from the result.


that resulted in (x + 3.25)^2 = x^2 + 6.5x + 3.25^2 -3.25^2 = x^2 + 6.5x


in other words, x^2 + 6.5x is equal to (x + 3.25)^2 - 3.25^2


so i wound up with x^2 + 6.5x = 7.5 becoming (x + 3.25)^2 - 3.25^2 = 7.5


the rest was solving for x as i did above.