Mister Exam
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Identical expressions
- sqrt(fourteen + five *x)=x
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Similar expressions
- sqrt(14-5*x)=x
sqrt(14+5*x)=x equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sqrt{5 x + 14} = x$$
$$\sqrt{5 x + 14} = x$$
We raise the equation sides to 2-th degree
$$5 x + 14 = x^{2}$$
$$5 x + 14 = x^{2}$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 5 x + 14 = 0$$
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
$$a = -1$$
$$b = 5$$
$$c = 14$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = -2$$
$$x_{2} = 7$$
Because
$$\sqrt{5 x + 14} = x$$
and
$$\sqrt{5 x + 14} \geq 0$$
then
$$x \geq 0$$
or
$$0 \leq x$$
$$x < \infty$$
The final answer:
$$x_{2} = 7$$
$$\sqrt{5 x + 14} = x$$
$$\sqrt{5 x + 14} = x$$
We raise the equation sides to 2-th degree
$$5 x + 14 = x^{2}$$
$$5 x + 14 = x^{2}$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 5 x + 14 = 0$$
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
$$a = -1$$
$$b = 5$$
$$c = 14$$
, then
D = b^2 - 4 * a * c =
(5)^2 - 4 * (-1) * (14) = 81
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = -2$$
$$x_{2} = 7$$
Because
$$\sqrt{5 x + 14} = x$$
and
$$\sqrt{5 x + 14} \geq 0$$
then
$$x \geq 0$$
or
$$0 \leq x$$
$$x < \infty$$
The final answer:
$$x_{2} = 7$$
The graph