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- 0.057*x^2-1.41x+3.52=0
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0.057*x^2-1.41x-3.52=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 57*x 141*x 88 ----- - ----- - -- = 0 1000 100 25
$$\frac{57 x^{2}}{1000} - \frac{141 x}{100} - \frac{88}{25} = 0$$
Detail solution
Expand the expression in the equation
$$\left(\frac{57 x^{2}}{1000} - \frac{141 x}{100} - \frac{88}{25}\right) + 0 = 0$$
We get the quadratic equation
$$\frac{57 x^{2}}{1000} - \frac{141 x}{100} - \frac{88}{25} = 0$$
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
$$a = \frac{57}{1000}$$
$$b = - \frac{141}{100}$$
$$c = - \frac{88}{25}$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{235}{19} + \frac{\sqrt{697665}}{57}$$
Simplify
$$x_{2} = - \frac{\sqrt{697665}}{57} + \frac{235}{19}$$
Simplify
$$\left(\frac{57 x^{2}}{1000} - \frac{141 x}{100} - \frac{88}{25}\right) + 0 = 0$$
We get the quadratic equation
$$\frac{57 x^{2}}{1000} - \frac{141 x}{100} - \frac{88}{25} = 0$$
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
$$a = \frac{57}{1000}$$
$$b = - \frac{141}{100}$$
$$c = - \frac{88}{25}$$
, then
D = b^2 - 4 * a * c =
(-141/100)^2 - 4 * (57/1000) * (-88/25) = 139533/50000
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{235}{19} + \frac{\sqrt{697665}}{57}$$
Simplify
$$x_{2} = - \frac{\sqrt{697665}}{57} + \frac{235}{19}$$
Simplify
Vieta's Theorem
rewrite the equation
$$\frac{57 x^{2}}{1000} - \frac{141 x}{100} - \frac{88}{25} = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{470 x}{19} - \frac{3520}{57} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{470}{19}$$
$$q = \frac{c}{a}$$
$$q = - \frac{3520}{57}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{470}{19}$$
$$x_{1} x_{2} = - \frac{3520}{57}$$
$$\frac{57 x^{2}}{1000} - \frac{141 x}{100} - \frac{88}{25} = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{470 x}{19} - \frac{3520}{57} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{470}{19}$$
$$q = \frac{c}{a}$$
$$q = - \frac{3520}{57}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{470}{19}$$
$$x_{1} x_{2} = - \frac{3520}{57}$$
Rapid solution
[src]
________
235 \/ 697665
x1 = --- - ----------
19 57
$$x_{1} = - \frac{\sqrt{697665}}{57} + \frac{235}{19}$$
________
235 \/ 697665
x2 = --- + ----------
19 57
$$x_{2} = \frac{235}{19} + \frac{\sqrt{697665}}{57}$$
Sum and product of roots
[src]
sum
________ ________
235 \/ 697665 235 \/ 697665
0 + --- - ---------- + --- + ----------
19 57 19 57
$$\left(\left(- \frac{\sqrt{697665}}{57} + \frac{235}{19}\right) + 0\right) + \left(\frac{235}{19} + \frac{\sqrt{697665}}{57}\right)$$
=
470 --- 19
$$\frac{470}{19}$$
product
/ ________\ / ________\ |235 \/ 697665 | |235 \/ 697665 | 1*|--- - ----------|*|--- + ----------| \ 19 57 / \ 19 57 /
$$1 \cdot \left(- \frac{\sqrt{697665}}{57} + \frac{235}{19}\right) \left(\frac{235}{19} + \frac{\sqrt{697665}}{57}\right)$$
=
-3520 ------ 57
$$- \frac{3520}{57}$$
-3520/57
The graph