First, expand the terms within parenthesis on both sides of the equation. Be careful to manage the signs correcly:
#(2ax * x) - (2ax * 3b) = 9 - 5a + 6b#
#2ax^2 - 6abx = 9 - 5a + 6b#
Next, put the equation in standard form:
#2ax^2 - 6abx - color(red)(9) + color(blue)(5a) - color(green)(6b) = 9 - color(red)(9) - 5a + color(blue)(5a) + 6b - color(green)(6b)#
#2ax^2 - 6abx - 9 + 5a - 6b = 0 - 0 + 0#
#2ax^2 - 6abx + (-9 + 5a - 6b) = 0#
We can use the quadratic equation to solve this problem for #x#:
The quadratic formula states:
For #color(red)(a)x^2 + color(blue)(b)x + color(green)(c) = 0#, the values of #x# which are the solutions to the equation are given by:
#x = (-color(blue)(b) +- sqrt(color(blue)(b)^2 - (4color(red)(a)color(green)(c))))/(2 * color(red)(a))#
Substituting:
#color(red)(2a)# for #color(red)(a)#
#color(blue)(-6ab)# for #color(blue)(b)#
#color(green)(-9 + 5a - 6b)# for #color(green)(c)# gives:
#x = (-color(blue)((-6ab)) +- sqrt(color(blue)((-6ab))^2 - (4 * color(red)(2a) * color(green)((-9 + 5a - 6b)))))/(2 * color(red)(2a))#
#x = (color(blue)(6ab) +- sqrt(36a^2b^2- (8acolor(green)((-9 + 5a - 6b)))))/(4a)#
#x = (color(blue)(6ab) +- sqrt(36a^2b^2- (-72a + 40a^2 - 48ab)))/(4a)#
#x = (color(blue)(6ab) +- sqrt(36a^2b^2 + 72a - 40a^2 + 48ab))/(4a)#
#x = (color(blue)(6ab))/(4a) +- (sqrt(4(9a^2b^2 + 18a - 10a^2 + 12ab)))/(4a)#
#x = (3b)/2 +- (sqrt(4)sqrt((9a^2b^2 + 18a - 10a^2 + 12ab)))/(4a)#
#x = (3b)/2 +- (2sqrt((9a^2b^2 + 18a - 10a^2 + 12ab)))/(4a)#
#x = (3b)/2 +- (sqrt((9a^2b^2 + 18a - 10a^2 + 12ab)))/(2a)#
