Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation cos(x)=√3/2
- Equation x^2-x-56=0
-
Equation x^3=-8
-
Equation 4x^2-12x+9=0
- Express {x} in terms of y where:
- 16*x-15*y=5
- 17*x-1*y=-15
- -8*x+12*y=-6
- -10*x-10*y=11
- Graphing y =:
- x+5
- Sum of series:
-
x+5
- Derivative of:
-
x+5
-
Identical expressions
- sqrt(4x+ seventeen)=x+ five
- square root of (4x plus 17) equally x plus 5
- square root of (4x plus seventeen) equally x plus five
- √(4x+17)=x+5
- sqrt4x+17=x+5
-
Similar expressions
- sqrt(4x-17)=x+5
- sqrt(4x+17)=x-5
sqrt(4x+17)=x+5 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sqrt{4 x + 17} = x + 5$$
$$\sqrt{4 x + 17} = x + 5$$
We raise the equation sides to 2-th degree
$$4 x + 17 = \left(x + 5\right)^{2}$$
$$4 x + 17 = x^{2} + 10 x + 25$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} - 6 x - 8 = 0$$
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
$$a = -1$$
$$b = -6$$
$$c = -8$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = -4$$
$$x_{2} = -2$$
Because
$$\sqrt{4 x + 17} = x + 5$$
and
$$\sqrt{4 x + 17} \geq 0$$
then
$$x + 5 \geq 0$$
or
$$-5 \leq x$$
$$x < \infty$$
The final answer:
$$x_{1} = -4$$
$$x_{2} = -2$$
$$\sqrt{4 x + 17} = x + 5$$
$$\sqrt{4 x + 17} = x + 5$$
We raise the equation sides to 2-th degree
$$4 x + 17 = \left(x + 5\right)^{2}$$
$$4 x + 17 = x^{2} + 10 x + 25$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} - 6 x - 8 = 0$$
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
$$a = -1$$
$$b = -6$$
$$c = -8$$
, then
D = b^2 - 4 * a * c =
(-6)^2 - 4 * (-1) * (-8) = 4
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = -4$$
$$x_{2} = -2$$
Because
$$\sqrt{4 x + 17} = x + 5$$
and
$$\sqrt{4 x + 17} \geq 0$$
then
$$x + 5 \geq 0$$
or
$$-5 \leq x$$
$$x < \infty$$
The final answer:
$$x_{1} = -4$$
$$x_{2} = -2$$
Sum and product of roots
[src]
sum
-4 - 2
$$-4 - 2$$
=
-6
$$-6$$
product
-4*(-2)
$$- -8$$
=
8
$$8$$
8