Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation -7x=4
-
Equation x^2-6=0
- Equation x^2-x-90=0
-
Equation x^2+7*x+12=0
- Express {x} in terms of y where:
- -6*x-11*y=-20
- -15*x+7*y=1
- -1*x-20*y=1
- -1*x-15*y=13
- Integral of d{x}:
-
sqrt(3x+1)
- Derivative of:
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sqrt(3x+1)
-
9-x
- Graphing y =:
- sqrt(3x+1)
- Limit of the function:
-
9-x
-
Identical expressions
- sqrt(3x+ one)= nine -x
- square root of (3x plus 1) equally 9 minus x
- square root of (3x plus one) equally nine minus x
- √(3x+1)=9-x
- sqrt3x+1=9-x
-
Similar expressions
- sqrt(3x+1)=9+x
- sqrt(3x-1)=9-x
- (x-3)(x^2+x+9)-x(x-1)(x+1)=10x
- sqrt(x)^2-4*x-12+sqrt(3*x)+12=0
- sqrt(3*x+15)=0
- sqrt3x+1=-9
sqrt(3x+1)=9-x equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sqrt{3 x + 1} = 9 - x$$
$$\sqrt{3 x + 1} = 9 - x$$
We raise the equation sides to 2-th degree
$$3 x + 1 = \left(9 - x\right)^{2}$$
$$3 x + 1 = x^{2} - 18 x + 81$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 21 x - 80 = 0$$
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
$$a = -1$$
$$b = 21$$
$$c = -80$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 5$$
$$x_{2} = 16$$
Because
$$\sqrt{3 x + 1} = 9 - x$$
and
$$\sqrt{3 x + 1} \geq 0$$
then
$$9 - x \geq 0$$
or
$$x \leq 9$$
$$-\infty < x$$
The final answer:
$$x_{1} = 5$$
$$\sqrt{3 x + 1} = 9 - x$$
$$\sqrt{3 x + 1} = 9 - x$$
We raise the equation sides to 2-th degree
$$3 x + 1 = \left(9 - x\right)^{2}$$
$$3 x + 1 = x^{2} - 18 x + 81$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + 21 x - 80 = 0$$
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
$$a = -1$$
$$b = 21$$
$$c = -80$$
, then
D = b^2 - 4 * a * c =
(21)^2 - 4 * (-1) * (-80) = 121
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 5$$
$$x_{2} = 16$$
Because
$$\sqrt{3 x + 1} = 9 - x$$
and
$$\sqrt{3 x + 1} \geq 0$$
then
$$9 - x \geq 0$$
or
$$x \leq 9$$
$$-\infty < x$$
The final answer:
$$x_{1} = 5$$