Mister Exam
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(3*x-2)*(3*x+2)+(4*x-5)^2=10*x+21 equation
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The solution
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2 (3*x - 2)*(3*x + 2) + (4*x - 5) = 10*x + 21
$$\left(3 x - 2\right) \left(3 x + 2\right) + \left(4 x - 5\right)^{2} = 10 x + 21$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(3 x - 2\right) \left(3 x + 2\right) + \left(4 x - 5\right)^{2} = 10 x + 21$$
to
$$\left(- 10 x - 21\right) + \left(\left(3 x - 2\right) \left(3 x + 2\right) + \left(4 x - 5\right)^{2}\right) = 0$$
Expand the expression in the equation
$$\left(- 10 x - 21\right) + \left(\left(3 x - 2\right) \left(3 x + 2\right) + \left(4 x - 5\right)^{2}\right) = 0$$
We get the quadratic equation
$$25 x^{2} - 50 x = 0$$
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
$$a = 25$$
$$b = -50$$
$$c = 0$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 2$$
$$x_{2} = 0$$
left part with negative sign.
The equation is transformed from
$$\left(3 x - 2\right) \left(3 x + 2\right) + \left(4 x - 5\right)^{2} = 10 x + 21$$
to
$$\left(- 10 x - 21\right) + \left(\left(3 x - 2\right) \left(3 x + 2\right) + \left(4 x - 5\right)^{2}\right) = 0$$
Expand the expression in the equation
$$\left(- 10 x - 21\right) + \left(\left(3 x - 2\right) \left(3 x + 2\right) + \left(4 x - 5\right)^{2}\right) = 0$$
We get the quadratic equation
$$25 x^{2} - 50 x = 0$$
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
$$a = 25$$
$$b = -50$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(-50)^2 - 4 * (25) * (0) = 2500
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 2$$
$$x_{2} = 0$$