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Identical expressions
- thirty *x^ two + fifteen , ninety-seven *x- one thousand, five hundred and ninety-seven = zero
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Similar expressions
- 30*x^2-15,97*x-1597=0
- 30*x^2+15,97*x+1597=0
30*x^2+15,97*x-1597=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 1597*x
30*x + ------ - 1597 = 0
100
$$\left(30 x^{2} + \frac{1597 x}{100}\right) - 1597 = 0$$
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
$$a = 30$$
$$b = \frac{1597}{100}$$
$$c = -1597$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{1597}{6000} + \frac{\sqrt{1918950409}}{6000}$$
$$x_{2} = - \frac{\sqrt{1918950409}}{6000} - \frac{1597}{6000}$$
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
$$a = 30$$
$$b = \frac{1597}{100}$$
$$c = -1597$$
, then
D = b^2 - 4 * a * c =
(1597/100)^2 - 4 * (30) * (-1597) = 1918950409/10000
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{1597}{6000} + \frac{\sqrt{1918950409}}{6000}$$
$$x_{2} = - \frac{\sqrt{1918950409}}{6000} - \frac{1597}{6000}$$
Vieta's Theorem
rewrite the equation
$$\left(30 x^{2} + \frac{1597 x}{100}\right) - 1597 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{1597 x}{3000} - \frac{1597}{30} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{1597}{3000}$$
$$q = \frac{c}{a}$$
$$q = - \frac{1597}{30}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{1597}{3000}$$
$$x_{1} x_{2} = - \frac{1597}{30}$$
$$\left(30 x^{2} + \frac{1597 x}{100}\right) - 1597 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{1597 x}{3000} - \frac{1597}{30} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{1597}{3000}$$
$$q = \frac{c}{a}$$
$$q = - \frac{1597}{30}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{1597}{3000}$$
$$x_{1} x_{2} = - \frac{1597}{30}$$
Rapid solution
[src]
____________
1597 \/ 1918950409
x1 = - ---- + --------------
6000 6000
$$x_{1} = - \frac{1597}{6000} + \frac{\sqrt{1918950409}}{6000}$$
____________
1597 \/ 1918950409
x2 = - ---- - --------------
6000 6000
$$x_{2} = - \frac{\sqrt{1918950409}}{6000} - \frac{1597}{6000}$$
x2 = -sqrt(1918950409)/6000 - 1597/6000
Sum and product of roots
[src]
sum
____________ ____________ 1597 \/ 1918950409 1597 \/ 1918950409 - ---- + -------------- + - ---- - -------------- 6000 6000 6000 6000
$$\left(- \frac{\sqrt{1918950409}}{6000} - \frac{1597}{6000}\right) + \left(- \frac{1597}{6000} + \frac{\sqrt{1918950409}}{6000}\right)$$
=
-1597 ------ 3000
$$- \frac{1597}{3000}$$
product
/ ____________\ / ____________\ | 1597 \/ 1918950409 | | 1597 \/ 1918950409 | |- ---- + --------------|*|- ---- - --------------| \ 6000 6000 / \ 6000 6000 /
$$\left(- \frac{1597}{6000} + \frac{\sqrt{1918950409}}{6000}\right) \left(- \frac{\sqrt{1918950409}}{6000} - \frac{1597}{6000}\right)$$
=
-1597 ------ 30
$$- \frac{1597}{30}$$
-1597/30