First, multiply each side of the equation by color(red)(12)12 to eliminate the fractions:
color(red)(12)(2/3x^2 + 1/4x) = color(red)(12) xx 312(23x2+14x)=12×3
(color(red)(12) xx 2/3x^2) + (color(red)(12) xx 1/4x) = 36(12×23x2)+(12×14x)=36
8x^2 + 3x = 368x2+3x=36
Next, put the equation in standard quadratic form:
8x^2 + 3x - color(red)(36) = 36 - color(red)(36)8x2+3x−36=36−36
8x^2 + 3x - 36 = 08x2+3x−36=0
We can nowuse the quadratic equation to solve this problem:
The quadratic formula states:
For color(red)(a)x^2 + color(blue)(b)x + color(green)(c) = 0ax2+bx+c=0, the values of xx 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))x=−b±√b2−(4ac)2⋅a
Substituting:
color(red)(8)8 for color(red)(a)a
color(blue)(3)3 for color(blue)(b)b
color(green)(-36)−36 for color(green)(c)c gives:
x = (-color(blue)(3) +- sqrt(color(blue)(3)^2 - (4 * color(red)(8) * color(green)(-36))))/(2 * color(red)(8))x=−3±√32−(4⋅8⋅−36)2⋅8
x = (-color(blue)(3) +- sqrt(9 - (-1152)))/16x=−3±√9−(−1152)16
x = (-color(blue)(3) +- sqrt(9 + 1152))/16x=−3±√9+115216
x = (-color(blue)(3) +- sqrt(1161))/16x=−3±√116116
x = (-color(blue)(3) +- sqrt(9 * 129))/16x=−3±√9⋅12916
x = (-color(blue)(3) +- sqrt(9)sqrt(129))/16x=−3±√9√12916
x = (-color(blue)(3) +- 3sqrt(129))/16x=−3±3√12916
x = (3(-1 +- sqrt(129)))/16x=3(−1±√129)16
Or
x = 3/16(-1 +- sqrt(129))x=316(−1±√129)
