Mister Exam
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Similar expressions
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sqrt(2*x+1)+sqrt(x-3)=4 equation
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The solution
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_________ _______ \/ 2*x + 1 + \/ x - 3 = 4
$$\sqrt{x - 3} + \sqrt{2 x + 1} = 4$$
Detail solution
Given the equation
$$\sqrt{x - 3} + \sqrt{2 x + 1} = 4$$
We raise the equation sides to 2-th degree
$$\left(\sqrt{x - 3} + \sqrt{2 x + 1}\right)^{2} = 16$$
or
$$1^{2} \left(2 x + 1\right) + \left(2 \sqrt{\left(x - 3\right) \left(2 x + 1\right)} + 1^{2} \left(x - 3\right)\right) = 16$$
or
$$3 x + 2 \sqrt{2 x^{2} - 5 x - 3} - 2 = 16$$
transform:
$$2 \sqrt{2 x^{2} - 5 x - 3} = 18 - 3 x$$
We raise the equation sides to 2-th degree
$$8 x^{2} - 20 x - 12 = \left(18 - 3 x\right)^{2}$$
$$8 x^{2} - 20 x - 12 = 9 x^{2} - 108 x + 324$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 88 x - 336 = 0$$
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
$$a = -1$$
$$b = 88$$
$$c = -336$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 4$$
$$x_{2} = 84$$
Because
$$\sqrt{2 x^{2} - 5 x - 3} = 9 - \frac{3 x}{2}$$
and
$$\sqrt{2 x^{2} - 5 x - 3} \geq 0$$
then
$$9 - \frac{3 x}{2} \geq 0$$
or
$$x \leq 6$$
$$-\infty < x$$
$$x_{1} = 4$$
check:
$$x_{1} = 4$$
$$\sqrt{x_{1} - 3} + \sqrt{2 x_{1} + 1} - 4 = 0$$
=
$$-4 + \left(\sqrt{-3 + 4} + \sqrt{1 + 2 \cdot 4}\right) = 0$$
=
- the identity
The final answer:
$$x_{1} = 4$$
$$\sqrt{x - 3} + \sqrt{2 x + 1} = 4$$
We raise the equation sides to 2-th degree
$$\left(\sqrt{x - 3} + \sqrt{2 x + 1}\right)^{2} = 16$$
or
$$1^{2} \left(2 x + 1\right) + \left(2 \sqrt{\left(x - 3\right) \left(2 x + 1\right)} + 1^{2} \left(x - 3\right)\right) = 16$$
or
$$3 x + 2 \sqrt{2 x^{2} - 5 x - 3} - 2 = 16$$
transform:
$$2 \sqrt{2 x^{2} - 5 x - 3} = 18 - 3 x$$
We raise the equation sides to 2-th degree
$$8 x^{2} - 20 x - 12 = \left(18 - 3 x\right)^{2}$$
$$8 x^{2} - 20 x - 12 = 9 x^{2} - 108 x + 324$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 88 x - 336 = 0$$
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
$$a = -1$$
$$b = 88$$
$$c = -336$$
, then
D = b^2 - 4 * a * c =
(88)^2 - 4 * (-1) * (-336) = 6400
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 4$$
$$x_{2} = 84$$
Because
$$\sqrt{2 x^{2} - 5 x - 3} = 9 - \frac{3 x}{2}$$
and
$$\sqrt{2 x^{2} - 5 x - 3} \geq 0$$
then
$$9 - \frac{3 x}{2} \geq 0$$
or
$$x \leq 6$$
$$-\infty < x$$
$$x_{1} = 4$$
check:
$$x_{1} = 4$$
$$\sqrt{x_{1} - 3} + \sqrt{2 x_{1} + 1} - 4 = 0$$
=
$$-4 + \left(\sqrt{-3 + 4} + \sqrt{1 + 2 \cdot 4}\right) = 0$$
=
0 = 0
- the identity
The final answer:
$$x_{1} = 4$$