Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation cos(x)=sqrt(3)/2
-
Equation x^2-8x+15=0
-
Equation 5-2*(x-1)=4-x
-
Equation (x-2)^2=(x-9)^2
- Express {x} in terms of y where:
- -6*x-2*y=-10
- -10*x+9*y=-6
- -14*x-16*y=14
- 20*x-4*y=-7
-
Identical expressions
- 3x^ two - twenty-seven = zero
- 3x squared minus 27 equally 0
- 3x to the power of two minus twenty minus seven equally zero
- 3x2-27=0
- 3x²-27=0
- 3x to the power of 2-27=0
- 3x^2-27=O
-
Similar expressions
- 3x^2+27=0
3x^2-27=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 3$$
$$b = 0$$
$$c = -27$$
, then
$$D = b^2 - 4\ a\ c = $$
$$0^{2} - 3 \cdot 4 \left(-27\right) = 324$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = 3$$
Simplify
$$x_{2} = -3$$
Simplify
$$a\ x^2 + b\ x + c = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 3$$
$$b = 0$$
$$c = -27$$
, then
$$D = b^2 - 4\ a\ c = $$
$$0^{2} - 3 \cdot 4 \left(-27\right) = 324$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = 3$$
Simplify
$$x_{2} = -3$$
Simplify
Vieta's Theorem
rewrite the equation
$$3 x^{2} - 27 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 9 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -9$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = -9$$
$$3 x^{2} - 27 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 9 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -9$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = -9$$
Sum and product of roots
[src]
sum
-3 + 3
$$\left(-3\right) + \left(3\right)$$
=
0
$$0$$
product
-3 * 3
$$\left(-3\right) * \left(3\right)$$
=
-9
$$-9$$
The graph