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16x^{2}+8x+1=\left(4x\right)^{2}+\left(x+1\right)^{2}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+1\right)^{2}.
16x^{2}+8x+1=4^{2}x^{2}+\left(x+1\right)^{2}
Expand \left(4x\right)^{2}.
16x^{2}+8x+1=16x^{2}+\left(x+1\right)^{2}
Calculate 4 to the power of 2 and get 16.
16x^{2}+8x+1=16x^{2}+x^{2}+2x+1
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
16x^{2}+8x+1=17x^{2}+2x+1
Combine 16x^{2} and x^{2} to get 17x^{2}.
16x^{2}+8x+1-17x^{2}=2x+1
Subtract 17x^{2} from both sides.
-x^{2}+8x+1=2x+1
Combine 16x^{2} and -17x^{2} to get -x^{2}.
-x^{2}+8x+1-2x=1
Subtract 2x from both sides.
-x^{2}+6x+1=1
Combine 8x and -2x to get 6x.
-x^{2}+6x+1-1=0
Subtract 1 from both sides.
-x^{2}+6x=0
Subtract 1 from 1 to get 0.
x\left(-x+6\right)=0
Factor out x.
x=0 x=6
To find equation solutions, solve x=0 and -x+6=0.
16x^{2}+8x+1=\left(4x\right)^{2}+\left(x+1\right)^{2}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+1\right)^{2}.
16x^{2}+8x+1=4^{2}x^{2}+\left(x+1\right)^{2}
Expand \left(4x\right)^{2}.
16x^{2}+8x+1=16x^{2}+\left(x+1\right)^{2}
Calculate 4 to the power of 2 and get 16.
16x^{2}+8x+1=16x^{2}+x^{2}+2x+1
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
16x^{2}+8x+1=17x^{2}+2x+1
Combine 16x^{2} and x^{2} to get 17x^{2}.
16x^{2}+8x+1-17x^{2}=2x+1
Subtract 17x^{2} from both sides.
-x^{2}+8x+1=2x+1
Combine 16x^{2} and -17x^{2} to get -x^{2}.
-x^{2}+8x+1-2x=1
Subtract 2x from both sides.
-x^{2}+6x+1=1
Combine 8x and -2x to get 6x.
-x^{2}+6x+1-1=0
Subtract 1 from both sides.
-x^{2}+6x=0
Subtract 1 from 1 to get 0.
x=\frac{-6±\sqrt{6^{2}}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 6 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-6±6}{2\left(-1\right)}
Take the square root of 6^{2}.
x=\frac{-6±6}{-2}
Multiply 2 times -1.
x=\frac{0}{-2}
Now solve the equation x=\frac{-6±6}{-2} when ± is plus. Add -6 to 6.
x=0
Divide 0 by -2.
x=-\frac{12}{-2}
Now solve the equation x=\frac{-6±6}{-2} when ± is minus. Subtract 6 from -6.
x=6
Divide -12 by -2.
x=0 x=6
The equation is now solved.
16x^{2}+8x+1=\left(4x\right)^{2}+\left(x+1\right)^{2}
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+1\right)^{2}.
16x^{2}+8x+1=4^{2}x^{2}+\left(x+1\right)^{2}
Expand \left(4x\right)^{2}.
16x^{2}+8x+1=16x^{2}+\left(x+1\right)^{2}
Calculate 4 to the power of 2 and get 16.
16x^{2}+8x+1=16x^{2}+x^{2}+2x+1
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.
16x^{2}+8x+1=17x^{2}+2x+1
Combine 16x^{2} and x^{2} to get 17x^{2}.
16x^{2}+8x+1-17x^{2}=2x+1
Subtract 17x^{2} from both sides.
-x^{2}+8x+1=2x+1
Combine 16x^{2} and -17x^{2} to get -x^{2}.
-x^{2}+8x+1-2x=1
Subtract 2x from both sides.
-x^{2}+6x+1=1
Combine 8x and -2x to get 6x.
-x^{2}+6x=1-1
Subtract 1 from both sides.
-x^{2}+6x=0
Subtract 1 from 1 to get 0.
\frac{-x^{2}+6x}{-1}=\frac{0}{-1}
Divide both sides by -1.
x^{2}+\frac{6}{-1}x=\frac{0}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-6x=\frac{0}{-1}
Divide 6 by -1.
x^{2}-6x=0
Divide 0 by -1.
x^{2}-6x+\left(-3\right)^{2}=\left(-3\right)^{2}
Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-6x+9=9
Square -3.
\left(x-3\right)^{2}=9
Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-3\right)^{2}}=\sqrt{9}
Take the square root of both sides of the equation.
x-3=3 x-3=-3
Simplify.
x=6 x=0
Add 3 to both sides of the equation.