Mister Exam
(x+4)/(5*x+9)=(x+4)/(4*x-5) equation
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The solution
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x + 4 x + 4 ------- = ------- 5*x + 9 4*x - 5
$$\frac{x + 4}{5 x + 9} = \frac{x + 4}{4 x - 5}$$
Detail solution
Given the equation:
$$\frac{x + 4}{5 x + 9} = \frac{x + 4}{4 x - 5}$$
Multiply the equation sides by the denominators:
9 + 5*x and -5 + 4*x
we get:
$$\frac{\left(x + 4\right) \left(5 x + 9\right)}{5 x + 9} = \frac{\left(x + 4\right) \left(5 x + 9\right)}{4 x - 5}$$
$$x + 4 = \frac{\left(x + 4\right) \left(5 x + 9\right)}{4 x - 5}$$
$$\left(x + 4\right) \left(4 x - 5\right) = \frac{\left(x + 4\right) \left(5 x + 9\right)}{4 x - 5} \left(4 x - 5\right)$$
$$4 x^{2} + 11 x - 20 = 5 x^{2} + 29 x + 36$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$4 x^{2} + 11 x - 20 = 5 x^{2} + 29 x + 36$$
to
$$- x^{2} - 18 x - 56 = 0$$
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
$$a = -1$$
$$b = -18$$
$$c = -56$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = -14$$
$$x_{2} = -4$$
$$\frac{x + 4}{5 x + 9} = \frac{x + 4}{4 x - 5}$$
Multiply the equation sides by the denominators:
9 + 5*x and -5 + 4*x
we get:
$$\frac{\left(x + 4\right) \left(5 x + 9\right)}{5 x + 9} = \frac{\left(x + 4\right) \left(5 x + 9\right)}{4 x - 5}$$
$$x + 4 = \frac{\left(x + 4\right) \left(5 x + 9\right)}{4 x - 5}$$
$$\left(x + 4\right) \left(4 x - 5\right) = \frac{\left(x + 4\right) \left(5 x + 9\right)}{4 x - 5} \left(4 x - 5\right)$$
$$4 x^{2} + 11 x - 20 = 5 x^{2} + 29 x + 36$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$4 x^{2} + 11 x - 20 = 5 x^{2} + 29 x + 36$$
to
$$- x^{2} - 18 x - 56 = 0$$
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
$$a = -1$$
$$b = -18$$
$$c = -56$$
, then
D = b^2 - 4 * a * c =
(-18)^2 - 4 * (-1) * (-56) = 100
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = -14$$
$$x_{2} = -4$$
Sum and product of roots
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sum
-14 - 4
$$-14 - 4$$
=
-18
$$-18$$
product
-14*(-4)
$$- -56$$
=
56
$$56$$
56