First, this how I interpreted your equaton.
2*4sqrt(5x^2 - 8x + 1) + 4 =33*2sqrt(5x^2 - 8x), solve for x
Second, this is a very long and complicated solution, but it is ACCURATE!
Solve for x:
4 + 8 sqrt(5 x^2 - 8 x + 1) = 66 sqrt(5 x^2 - 8 x)
Subtract 4 from both sides:
8 sqrt(5 x^2 - 8 x + 1) = 66 sqrt(5 x^2 - 8 x) - 4
Raise both sides to the power of two:
64 (5 x^2 - 8 x + 1) = (66 sqrt(5 x^2 - 8 x) - 4)^2
Expand out terms of the left hand side:
320 x^2 - 512 x + 64 = (66 sqrt(5 x^2 - 8 x) - 4)^2
(66 sqrt(5 x^2 - 8 x) - 4)^2 = 16 - 34848 x + 21780 x^2 - 528 sqrt(5 x^2 - 8 x):
320 x^2 - 512 x + 64 = 16 - 34848 x + 21780 x^2 - 528 sqrt(5 x^2 - 8 x)
Subtract 64 - 512 x + 320 x^2 - 528 sqrt(5 x^2 - 8 x) from both sides:
528 sqrt(5 x^2 - 8 x) = 21460 x^2 - 34336 x - 48
Raise both sides to the power of two:
278784 (5 x^2 - 8 x) = (21460 x^2 - 34336 x - 48)^2
Expand out terms of the left hand side:
1393920 x^2 - 2230272 x = (21460 x^2 - 34336 x - 48)^2
Expand out terms of the right hand side:
1393920 x^2 - 2230272 x = 460531600 x^4 - 1473701120 x^3 + 1176900736 x^2 + 3296256 x + 2304
Subtract 460531600 x^4 - 1473701120 x^3 + 1176900736 x^2 + 3296256 x + 2304 from both sides:
-460531600 x^4 + 1473701120 x^3 - 1175506816 x^2 - 5526528 x - 2304 = 0
Factor constant terms from the left hand side:
-16 (28783225 x^4 - 92106320 x^3 + 73469176 x^2 + 345408 x + 144) = 0
Divide both sides by -16:
28783225 x^4 - 92106320 x^3 + 73469176 x^2 + 345408 x + 144 = 0
Eliminate the cubic term by substituting y = x - 4/5:
144 + 345408 (y + 4/5) + 73469176 (y + 4/5)^2 - 92106320 (y + 4/5)^3 + 28783225 (y + 4/5)^4 = 0
Expand out terms of the left hand side:
28783225 y^4 - 37058408 y^2 + 298197904/25 = 0
Substitute z = y^2:
28783225 z^2 - 37058408 z + 298197904/25 = 0
Divide both sides by 28783225:
z^2 - (37058408 z)/28783225 + 298197904/719580625 = 0
Subtract 298197904/719580625 from both sides:
z^2 - (37058408 z)/28783225 = -298197904/719580625
Add 343331400873616/828474041400625 to both sides:
z^2 - (37058408 z)/28783225 + 343331400873616/828474041400625 = 300250368/33138961656025
Write the left hand side as a square:
(z - 18529204/28783225)^2 = 300250368/33138961656025
Take the square root of both sides:
z - 18529204/28783225 = (528 sqrt(1077))/5756645 or z - 18529204/28783225 = -(528 sqrt(1077))/5756645
Add 18529204/28783225 to both sides:
z = 18529204/28783225 + (528 sqrt(1077))/5756645 or z - 18529204/28783225 = -(528 sqrt(1077))/5756645
Substitute back for z = y^2:
y^2 = 18529204/28783225 + (528 sqrt(1077))/5756645 or z - 18529204/28783225 = -(528 sqrt(1077))/5756645
Take the square root of both sides:
y = sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or y = -sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or z - 18529204/28783225 = -(528 sqrt(1077))/5756645
Substitute back for y = x - 4/5:
x - 4/5 = sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or y = -sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or z - 18529204/28783225 = -(528 sqrt(1077))/5756645
Add 4/5 to both sides:
x = 4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or y = -sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or z - 18529204/28783225 = -(528 sqrt(1077))/5756645
Substitute back for y = x - 4/5:
x = 4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x - 4/5 = -sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or z - 18529204/28783225 = -(528 sqrt(1077))/5756645
Add 4/5 to both sides:
x = 4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x = 4/5 - sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or z - 18529204/28783225 = -(528 sqrt(1077))/5756645
Add 18529204/28783225 to both sides:
x = 4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x = 4/5 - sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or z = 18529204/28783225 - (528 sqrt(1077))/5756645
Substitute back for z = y^2:
x = 4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x = 4/5 - sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or y^2 = 18529204/28783225 - (528 sqrt(1077))/5756645
Take the square root of both sides:
x = 4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x = 4/5 - sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or y = sqrt(18529204/28783225 - (528 sqrt(1077))/5756645) or y = -sqrt(18529204/28783225 - (528 sqrt(1077))/5756645)
Substitute back for y = x - 4/5:
x = 4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x = 4/5 - sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x - 4/5 = sqrt(18529204/28783225 - (528 sqrt(1077))/5756645) or y = -sqrt(18529204/28783225 - (528 sqrt(1077))/5756645)
Add 4/5 to both sides:
x = 4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x = 4/5 - sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x = 4/5 + sqrt(18529204/28783225 - (528 sqrt(1077))/5756645) or y = -sqrt(18529204/28783225 - (528 sqrt(1077))/5756645)
Substitute back for y = x - 4/5:
x = 4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x = 4/5 - sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x = 4/5 + sqrt(18529204/28783225 - (528 sqrt(1077))/5756645) or x - 4/5 = -sqrt(18529204/28783225 - (528 sqrt(1077))/5756645)
Add 4/5 to both sides:
x = 4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x = 4/5 - sqrt(18529204/28783225 + (528 sqrt(1077))/5756645) or x = 4/5 + sqrt(18529204/28783225 - (528 sqrt(1077))/5756645) or x = 4/5 - sqrt(18529204/28783225 - (528 sqrt(1077))/5756645)
4 + 8 sqrt(5 x^2 - 8 x + 1) ≈ 12.0148
66 sqrt(5 x^2 - 8 x) ≈ 4.01479:
So this solution is incorrect
4 + 8 sqrt(5 x^2 - 8 x + 1) ⇒ 4 + 8 sqrt(1 - 8 (4/5 + sqrt(18529204/28783225 - (528 sqrt(1077))/5756645)) + 5 (4/5 + sqrt(18529204/28783225 - (528 sqrt(1077))/5756645))^2) = 4 + (8 sqrt(1172917 - 528 sqrt(1077)))/1073 ≈ 12.0148
66 sqrt(5 x^2 - 8 x) ⇒ 66 sqrt(5 (4/5 + sqrt(18529204/28783225 - (528 sqrt(1077))/5756645))^2 - 8 (4/5 + sqrt(18529204/28783225 - (528 sqrt(1077))/5756645))) = (132 sqrt(5397 - 132 sqrt(1077)))/1073 ≈ 4.01479:
So this solution is incorrect
4 + 8 sqrt(5 x^2 - 8 x + 1) ≈ 12.1341
66 sqrt(5 x^2 - 8 x) ≈ 12.1341:
So this solution is correct
4 + 8 sqrt(5 x^2 - 8 x + 1) ⇒ 4 + 8 sqrt(1 - 8 (4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645)) + 5 (4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645))^2) = 4 + (8 sqrt(1172917 + 528 sqrt(1077)))/1073 ≈ 12.1341
66 sqrt(5 x^2 - 8 x) ⇒ 66 sqrt(5 (4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645))^2 - 8 (4/5 + sqrt(18529204/28783225 + (528 sqrt(1077))/5756645))) = (132 sqrt(5397 + 132 sqrt(1077)))/1073 ≈ 12.1341:
So this solution is correct
The solutions are:
Answer: |x = 4/5 + sqrt(18529204/28783225 + (528 sqr(1077))/5756645 or x = 4/5 - sqrt(18529204/28783225 + (528 sqrt(1077))/5756645)
I choose to read this as
\(\displaystyle 2.4^{\sqrt{(5x^{2}-8x)}+1}+4 = 33.2^{\sqrt{(5x^{2}-8x})}\).
Start by making a substitution, let \(\displaystyle z = \sqrt{5x^{2}-8x}\)
and the equation becomes
\(\displaystyle 2.4^{z+1}+4=33.2^{z}\).
Next write the first 4 as 2 squared and simplify,
\(\displaystyle 8.2^{2z}-33.2^{z}+4=0\).
That's a quadratic in 2^z, so let y = 2^z and we have
\(\displaystyle 8y^{2}-33y+4=0\),
\(\displaystyle (8y-1)(y-4)=0\),
so y = 1/8 or y = 4.
Now back substitute to find z and then back substitute to find x, (and you have to solve another quadratic to do that,
i.e \(\displaystyle 5x^{2}-8x=z^{2}\)).
You should find that x = -2/5 or x = 2.
Tiggsy.
+33653 Choosing to read the "dots" as multiplications, there are two real solutions:
+33653 These two solutions may be obtained by letting y = 2^sqrt(5x^2 - 8x) so the equation becomes:
8y^2 + 4 = 33y
(8y - 1)(y - 4) = 0
y = 1/8, or y = 4
y = 2^(-3) or y = 2^2
We can't have a square root being negative in the real number domain, so we must have:
sqrt(5x^2 - 8x) = 2
5x^2 - 8x = 4
5x^2 - 8x - 4 = 0
(5x + 2)(x - 2) = 0
x = -2/5 or x = 2
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