Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x^3+4*x^2-x-4=0
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Equation (x-7)^2=(9-x)^2
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Equation 3*x+50=77
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Equation 5-2(x-1)=4-x
- Express {x} in terms of y where:
- -19*x+1*y=0
- -12*x-3*y=-9
- 18*x-20*y=-7
- 18*x-12*y=-20
- Factor squared:
- x^2-4*x-6
-
Identical expressions
- x^ two - four *x- six = zero
- x squared minus 4 multiply by x minus 6 equally 0
- x to the power of two minus four multiply by x minus six equally zero
- x2-4*x-6=0
- x²-4*x-6=0
- x to the power of 2-4*x-6=0
- x^2-4x-6=0
- x2-4x-6=0
- x^2-4*x-6=O
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Similar expressions
- x^2-4*x+6=0
- x^2+4*x-6=0
x^2-4*x-6=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
$$a = 1$$
$$b = -4$$
$$c = -6$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-4\right)^{2} - 1 \cdot 4 \left(-6\right) = 40$$
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
$$x_{1} = 2 + \sqrt{10}$$
Simplify
$$x_{2} = - \sqrt{10} + 2$$
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
$$a = 1$$
$$b = -4$$
$$c = -6$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-4\right)^{2} - 1 \cdot 4 \left(-6\right) = 40$$
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
$$x_{1} = 2 + \sqrt{10}$$
Simplify
$$x_{2} = - \sqrt{10} + 2$$
Simplify
Vieta's Theorem
it is reduced quadratic equation
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -6$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = -6$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -6$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = -6$$
Rapid solution
[src]
____ x_1 = 2 - \/ 10
$$x_{1} = - \sqrt{10} + 2$$
____ x_2 = 2 + \/ 10
$$x_{2} = 2 + \sqrt{10}$$
Sum and product of roots
[src]
sum
____ ____ 2 - \/ 10 + 2 + \/ 10
$$\left(- \sqrt{10} + 2\right) + \left(2 + \sqrt{10}\right)$$
=
4
$$4$$
product
____ ____ 2 - \/ 10 * 2 + \/ 10
$$\left(- \sqrt{10} + 2\right) * \left(2 + \sqrt{10}\right)$$
=
-6
$$-6$$
The graph