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128(48-x)+128(48+x)=6(48-x)(48+x) equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
128*(48 - x) + 128*(48 + x) = 6*(48 - x)*(48 + x)
$$128 \left(48 - x\right) + 128 \left(x + 48\right) = 6 \left(48 - x\right) \left(x + 48\right)$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$128 \left(48 - x\right) + 128 \left(x + 48\right) = 6 \left(48 - x\right) \left(x + 48\right)$$
to
$$- 6 \left(48 - x\right) \left(x + 48\right) + \left(128 \left(48 - x\right) + 128 \left(x + 48\right)\right) = 0$$
Expand the expression in the equation
$$- 6 \left(48 - x\right) \left(x + 48\right) + \left(128 \left(48 - x\right) + 128 \left(x + 48\right)\right) = 0$$
We get the quadratic equation
$$6 x^{2} - 1536 = 0$$
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
$$a = 6$$
$$b = 0$$
$$c = -1536$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 16$$
$$x_{2} = -16$$
left part with negative sign.
The equation is transformed from
$$128 \left(48 - x\right) + 128 \left(x + 48\right) = 6 \left(48 - x\right) \left(x + 48\right)$$
to
$$- 6 \left(48 - x\right) \left(x + 48\right) + \left(128 \left(48 - x\right) + 128 \left(x + 48\right)\right) = 0$$
Expand the expression in the equation
$$- 6 \left(48 - x\right) \left(x + 48\right) + \left(128 \left(48 - x\right) + 128 \left(x + 48\right)\right) = 0$$
We get the quadratic equation
$$6 x^{2} - 1536 = 0$$
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
$$a = 6$$
$$b = 0$$
$$c = -1536$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (6) * (-1536) = 36864
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 16$$
$$x_{2} = -16$$
Sum and product of roots
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sum
-16 + 16
$$-16 + 16$$
=
0
$$0$$
product
-16*16
$$- 256$$
=
-256
$$-256$$
-256