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(3*x-16)/(7*x-3)+(9+3*x)/(7*x+4)=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
3*x - 16 9 + 3*x -------- + ------- = 1 7*x - 3 7*x + 4
$$\frac{3 x - 16}{7 x - 3} + \frac{3 x + 9}{7 x + 4} = 1$$
Detail solution
Given the equation:
$$\frac{3 x - 16}{7 x - 3} + \frac{3 x + 9}{7 x + 4} = 1$$
Multiply the equation sides by the denominators:
-3 + 7*x and 4 + 7*x
we get:
$$\left(7 x - 3\right) \left(\frac{3 x - 16}{7 x - 3} + \frac{3 x + 9}{7 x + 4}\right) = 7 x - 3$$
$$\frac{42 x^{2} - 46 x - 91}{7 x + 4} = 7 x - 3$$
$$\frac{42 x^{2} - 46 x - 91}{7 x + 4} \left(7 x + 4\right) = \left(7 x - 3\right) \left(7 x + 4\right)$$
$$42 x^{2} - 46 x - 91 = 49 x^{2} + 7 x - 12$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$42 x^{2} - 46 x - 91 = 49 x^{2} + 7 x - 12$$
to
$$- 7 x^{2} - 53 x - 79 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -7$$
$$b = -53$$
$$c = -79$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{53}{14} - \frac{\sqrt{597}}{14}$$
$$x_{2} = - \frac{53}{14} + \frac{\sqrt{597}}{14}$$
$$\frac{3 x - 16}{7 x - 3} + \frac{3 x + 9}{7 x + 4} = 1$$
Multiply the equation sides by the denominators:
-3 + 7*x and 4 + 7*x
we get:
$$\left(7 x - 3\right) \left(\frac{3 x - 16}{7 x - 3} + \frac{3 x + 9}{7 x + 4}\right) = 7 x - 3$$
$$\frac{42 x^{2} - 46 x - 91}{7 x + 4} = 7 x - 3$$
$$\frac{42 x^{2} - 46 x - 91}{7 x + 4} \left(7 x + 4\right) = \left(7 x - 3\right) \left(7 x + 4\right)$$
$$42 x^{2} - 46 x - 91 = 49 x^{2} + 7 x - 12$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$42 x^{2} - 46 x - 91 = 49 x^{2} + 7 x - 12$$
to
$$- 7 x^{2} - 53 x - 79 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -7$$
$$b = -53$$
$$c = -79$$
, then
D = b^2 - 4 * a * c =
(-53)^2 - 4 * (-7) * (-79) = 597
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{53}{14} - \frac{\sqrt{597}}{14}$$
$$x_{2} = - \frac{53}{14} + \frac{\sqrt{597}}{14}$$
Sum and product of roots
[src]
sum
_____ _____ 53 \/ 597 53 \/ 597 - -- - ------- + - -- + ------- 14 14 14 14
$$\left(- \frac{53}{14} - \frac{\sqrt{597}}{14}\right) + \left(- \frac{53}{14} + \frac{\sqrt{597}}{14}\right)$$
=
-53/7
$$- \frac{53}{7}$$
product
/ _____\ / _____\ | 53 \/ 597 | | 53 \/ 597 | |- -- - -------|*|- -- + -------| \ 14 14 / \ 14 14 /
$$\left(- \frac{53}{14} - \frac{\sqrt{597}}{14}\right) \left(- \frac{53}{14} + \frac{\sqrt{597}}{14}\right)$$
=
79/7
$$\frac{79}{7}$$
79/7
Rapid solution
[src]
_____
53 \/ 597
x1 = - -- - -------
14 14
$$x_{1} = - \frac{53}{14} - \frac{\sqrt{597}}{14}$$
_____
53 \/ 597
x2 = - -- + -------
14 14
$$x_{2} = - \frac{53}{14} + \frac{\sqrt{597}}{14}$$
x2 = -53/14 + sqrt(597)/14