Mister Exam
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x-sqrt(x)-12=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\left(- \sqrt{x} + x\right) - 12 = 0$$
$$- \sqrt{x} = 12 - x$$
We raise the equation sides to 2-th degree
$$x = \left(12 - x\right)^{2}$$
$$x = x^{2} - 24 x + 144$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 25 x - 144 = 0$$
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
$$a = -1$$
$$b = 25$$
$$c = -144$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 9$$
$$x_{2} = 16$$
Because
$$\sqrt{x} = x - 12$$
and
$$\sqrt{x} \geq 0$$
then
$$x - 12 \geq 0$$
or
$$12 \leq x$$
$$x < \infty$$
The final answer:
$$x_{2} = 16$$
$$\left(- \sqrt{x} + x\right) - 12 = 0$$
$$- \sqrt{x} = 12 - x$$
We raise the equation sides to 2-th degree
$$x = \left(12 - x\right)^{2}$$
$$x = x^{2} - 24 x + 144$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 25 x - 144 = 0$$
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
$$a = -1$$
$$b = 25$$
$$c = -144$$
, then
D = b^2 - 4 * a * c =
(25)^2 - 4 * (-1) * (-144) = 49
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 9$$
$$x_{2} = 16$$
Because
$$\sqrt{x} = x - 12$$
and
$$\sqrt{x} \geq 0$$
then
$$x - 12 \geq 0$$
or
$$12 \leq x$$
$$x < \infty$$
The final answer:
$$x_{2} = 16$$
The graph