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- Equation:
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Equation x^2+2*x+10=0
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Equation 2^4-2*x=64
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Equation x^2+3*x+9=0
- Equation x^2-x+30=0
- Express {x} in terms of y where:
- -18*x-3*y=2
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- 20*x-8*y=13
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Identical expressions
- x^ two + three *x- nine = zero
- x squared plus 3 multiply by x minus 9 equally 0
- x to the power of two plus three multiply by x minus nine equally zero
- x2+3*x-9=0
- x²+3*x-9=0
- x to the power of 2+3*x-9=0
- x^2+3x-9=0
- x2+3x-9=0
- x^2+3*x-9=O
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Similar expressions
- x^2-3*x-9=0
- x^2+3*x+9=0
x^2+3*x-9=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
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
$$a = 1$$
$$b = 3$$
$$c = -9$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{3}{2} + \frac{3 \sqrt{5}}{2}$$
$$x_{2} = - \frac{3 \sqrt{5}}{2} - \frac{3}{2}$$
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
$$a = 1$$
$$b = 3$$
$$c = -9$$
, then
D = b^2 - 4 * a * c =
(3)^2 - 4 * (1) * (-9) = 45
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{3}{2} + \frac{3 \sqrt{5}}{2}$$
$$x_{2} = - \frac{3 \sqrt{5}}{2} - \frac{3}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = -9$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3$$
$$x_{1} x_{2} = -9$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = -9$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3$$
$$x_{1} x_{2} = -9$$
Sum and product of roots
[src]
sum
___ ___ 3 3*\/ 5 3 3*\/ 5 - - + ------- + - - - ------- 2 2 2 2
$$\left(- \frac{3 \sqrt{5}}{2} - \frac{3}{2}\right) + \left(- \frac{3}{2} + \frac{3 \sqrt{5}}{2}\right)$$
=
-3
$$-3$$
product
/ ___\ / ___\ | 3 3*\/ 5 | | 3 3*\/ 5 | |- - + -------|*|- - - -------| \ 2 2 / \ 2 2 /
$$\left(- \frac{3}{2} + \frac{3 \sqrt{5}}{2}\right) \left(- \frac{3 \sqrt{5}}{2} - \frac{3}{2}\right)$$
=
-9
$$-9$$
-9
Rapid solution
[src]
___
3 3*\/ 5
x1 = - - + -------
2 2
$$x_{1} = - \frac{3}{2} + \frac{3 \sqrt{5}}{2}$$
___
3 3*\/ 5
x2 = - - - -------
2 2
$$x_{2} = - \frac{3 \sqrt{5}}{2} - \frac{3}{2}$$
x2 = -3*sqrt(5)/2 - 3/2
The graph