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Identical expressions
- (8y+ five)^ two - one hundred and twenty-eight =80y-64y^ two + twenty-five
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Similar expressions
- (8y+5)^2-128=80y+64y^2+25
- (8y-5)^2-128=80y-64y^2+25
- (8*y+5)^2-128=80*y-64^2+25
- (8y+5)^2-128=80y-64y^2-25
- (8y+5)^2+128=80y-64y^2+25
(8y+5)^2-128=80y-64y^2+25 equation
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The solution
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[src]
2 2 (8*y + 5) - 128 = 80*y - 64*y + 25
$$\left(8 y + 5\right)^{2} - 128 = - 64 y^{2} + 80 y + 25$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(8 y + 5\right)^{2} - 128 = - 64 y^{2} + 80 y + 25$$
to
$$\left(\left(8 y + 5\right)^{2} - 128\right) - \left(- 64 y^{2} + 80 y + 25\right) = 0$$
Expand the expression in the equation
$$\left(\left(8 y + 5\right)^{2} - 128\right) - \left(- 64 y^{2} + 80 y + 25\right) = 0$$
We get the quadratic equation
$$128 y^{2} - 128 = 0$$
This equation is of the form
$$a\ y^2 + b\ y + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 128$$
$$b = 0$$
$$c = -128$$
, then
$$D = b^2 - 4\ a\ c = $$
$$0^{2} - 128 \cdot 4 \left(-128\right) = 65536$$
Because D > 0, then the equation has two roots.
$$y_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$y_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$y_{1} = 1$$
Simplify
$$y_{2} = -1$$
Simplify
left part with negative sign.
The equation is transformed from
$$\left(8 y + 5\right)^{2} - 128 = - 64 y^{2} + 80 y + 25$$
to
$$\left(\left(8 y + 5\right)^{2} - 128\right) - \left(- 64 y^{2} + 80 y + 25\right) = 0$$
Expand the expression in the equation
$$\left(\left(8 y + 5\right)^{2} - 128\right) - \left(- 64 y^{2} + 80 y + 25\right) = 0$$
We get the quadratic equation
$$128 y^{2} - 128 = 0$$
This equation is of the form
$$a\ y^2 + b\ y + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 128$$
$$b = 0$$
$$c = -128$$
, then
$$D = b^2 - 4\ a\ c = $$
$$0^{2} - 128 \cdot 4 \left(-128\right) = 65536$$
Because D > 0, then the equation has two roots.
$$y_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$y_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$y_{1} = 1$$
Simplify
$$y_{2} = -1$$
Simplify
Sum and product of roots
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sum
-1 + 1
$$\left(-1\right) + \left(1\right)$$
=
0
$$0$$
product
-1 * 1
$$\left(-1\right) * \left(1\right)$$
=
-1
$$-1$$
The graph