Mister Exam
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How to use it?
- Equation:
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Equation (-2-4t-1)²+(0+2t-1)²+(2+5t)²=9
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Identical expressions
- 5x^ two -7x- two = zero
- 5x squared minus 7x minus 2 equally 0
- 5x to the power of two minus 7x minus two equally zero
- 5x2-7x-2=0
- 5x²-7x-2=0
- 5x to the power of 2-7x-2=0
- 5x^2-7x-2=O
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Similar expressions
- 5x^2+7x-2=0
- 5x^2-7x+2=0
5x^2-7x-2=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
$$a = 5$$
$$b = -7$$
$$c = -2$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{7}{10} + \frac{\sqrt{89}}{10}$$
$$x_{2} = \frac{7}{10} - \frac{\sqrt{89}}{10}$$
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
$$a = 5$$
$$b = -7$$
$$c = -2$$
, then
D = b^2 - 4 * a * c =
(-7)^2 - 4 * (5) * (-2) = 89
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{7}{10} + \frac{\sqrt{89}}{10}$$
$$x_{2} = \frac{7}{10} - \frac{\sqrt{89}}{10}$$
Vieta's Theorem
rewrite the equation
$$\left(5 x^{2} - 7 x\right) - 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{7 x}{5} - \frac{2}{5} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{7}{5}$$
$$q = \frac{c}{a}$$
$$q = - \frac{2}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{7}{5}$$
$$x_{1} x_{2} = - \frac{2}{5}$$
$$\left(5 x^{2} - 7 x\right) - 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{7 x}{5} - \frac{2}{5} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{7}{5}$$
$$q = \frac{c}{a}$$
$$q = - \frac{2}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{7}{5}$$
$$x_{1} x_{2} = - \frac{2}{5}$$
Sum and product of roots
[src]
sum
____ ____ 7 \/ 89 7 \/ 89 -- - ------ + -- + ------ 10 10 10 10
$$\left(\frac{7}{10} - \frac{\sqrt{89}}{10}\right) + \left(\frac{7}{10} + \frac{\sqrt{89}}{10}\right)$$
=
7/5
$$\frac{7}{5}$$
product
/ ____\ / ____\ |7 \/ 89 | |7 \/ 89 | |-- - ------|*|-- + ------| \10 10 / \10 10 /
$$\left(\frac{7}{10} - \frac{\sqrt{89}}{10}\right) \left(\frac{7}{10} + \frac{\sqrt{89}}{10}\right)$$
=
-2/5
$$- \frac{2}{5}$$
-2/5
Rapid solution
[src]
____
7 \/ 89
x1 = -- - ------
10 10
$$x_{1} = \frac{7}{10} - \frac{\sqrt{89}}{10}$$
____
7 \/ 89
x2 = -- + ------
10 10
$$x_{2} = \frac{7}{10} + \frac{\sqrt{89}}{10}$$
x2 = 7/10 + sqrt(89)/10