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\left(2.5\left(-4\right)x\right)^{2}=\left(1.5\sqrt{16x^{2}+4^{2}}\right)^{2}
Square both sides of the equation.
\left(-10x\right)^{2}=\left(1.5\sqrt{16x^{2}+4^{2}}\right)^{2}
Multiply 2.5 and -4 to get -10.
\left(-10\right)^{2}x^{2}=\left(1.5\sqrt{16x^{2}+4^{2}}\right)^{2}
Expand \left(-10x\right)^{2}.
100x^{2}=\left(1.5\sqrt{16x^{2}+4^{2}}\right)^{2}
Calculate -10 to the power of 2 and get 100.
100x^{2}=\left(1.5\sqrt{16x^{2}+16}\right)^{2}
Calculate 4 to the power of 2 and get 16.
100x^{2}=1.5^{2}\left(\sqrt{16x^{2}+16}\right)^{2}
Expand \left(1.5\sqrt{16x^{2}+16}\right)^{2}.
100x^{2}=2.25\left(\sqrt{16x^{2}+16}\right)^{2}
Calculate 1.5 to the power of 2 and get 2.25.
100x^{2}=2.25\left(16x^{2}+16\right)
Calculate \sqrt{16x^{2}+16} to the power of 2 and get 16x^{2}+16.
100x^{2}=36x^{2}+36
Use the distributive property to multiply 2.25 by 16x^{2}+16.
100x^{2}-36x^{2}=36
Subtract 36x^{2} from both sides.
64x^{2}=36
Combine 100x^{2} and -36x^{2} to get 64x^{2}.
64x^{2}-36=0
Subtract 36 from both sides.
16x^{2}-9=0
Divide both sides by 4.
\left(4x-3\right)\left(4x+3\right)=0
Consider 16x^{2}-9. Rewrite 16x^{2}-9 as \left(4x\right)^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).
x=\frac{3}{4} x=-\frac{3}{4}
To find equation solutions, solve 4x-3=0 and 4x+3=0.
2.5\left(-4\right)\times \frac{3}{4}=1.5\sqrt{16\times \left(\frac{3}{4}\right)^{2}+4^{2}}
Substitute \frac{3}{4} for x in the equation 2.5\left(-4\right)x=1.5\sqrt{16x^{2}+4^{2}}.
-7.5=7.5
Simplify. The value x=\frac{3}{4} does not satisfy the equation because the left and the right hand side have opposite signs.
2.5\left(-4\right)\left(-\frac{3}{4}\right)=1.5\sqrt{16\left(-\frac{3}{4}\right)^{2}+4^{2}}
Substitute -\frac{3}{4} for x in the equation 2.5\left(-4\right)x=1.5\sqrt{16x^{2}+4^{2}}.
7.5=7.5
Simplify. The value x=-\frac{3}{4} satisfies the equation.
x=-\frac{3}{4}
Equation -10x=\frac{3\sqrt{16x^{2}+16}}{2} has a unique solution.