Solve for x
x=16\sqrt{3}+32\approx 59.712812921
x=32-16\sqrt{3}\approx 4.287187079
Graph
Share
Copied to clipboard
\frac{1}{16}x^{2}-4x+16=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times \frac{1}{16}\times 16}}{2\times \frac{1}{16}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{16} for a, -4 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-4\right)±\sqrt{16-4\times \frac{1}{16}\times 16}}{2\times \frac{1}{16}}
Square -4.
x=\frac{-\left(-4\right)±\sqrt{16-\frac{1}{4}\times 16}}{2\times \frac{1}{16}}
Multiply -4 times \frac{1}{16}.
x=\frac{-\left(-4\right)±\sqrt{16-4}}{2\times \frac{1}{16}}
Multiply -\frac{1}{4} times 16.
x=\frac{-\left(-4\right)±\sqrt{12}}{2\times \frac{1}{16}}
Add 16 to -4.
x=\frac{-\left(-4\right)±2\sqrt{3}}{2\times \frac{1}{16}}
Take the square root of 12.
x=\frac{4±2\sqrt{3}}{2\times \frac{1}{16}}
The opposite of -4 is 4.
x=\frac{4±2\sqrt{3}}{\frac{1}{8}}
Multiply 2 times \frac{1}{16}.
x=\frac{2\sqrt{3}+4}{\frac{1}{8}}
Now solve the equation x=\frac{4±2\sqrt{3}}{\frac{1}{8}} when ± is plus. Add 4 to 2\sqrt{3}.
x=16\sqrt{3}+32
Divide 4+2\sqrt{3} by \frac{1}{8} by multiplying 4+2\sqrt{3} by the reciprocal of \frac{1}{8}.
x=\frac{4-2\sqrt{3}}{\frac{1}{8}}
Now solve the equation x=\frac{4±2\sqrt{3}}{\frac{1}{8}} when ± is minus. Subtract 2\sqrt{3} from 4.
x=32-16\sqrt{3}
Divide 4-2\sqrt{3} by \frac{1}{8} by multiplying 4-2\sqrt{3} by the reciprocal of \frac{1}{8}.
x=16\sqrt{3}+32 x=32-16\sqrt{3}
The equation is now solved.
\frac{1}{16}x^{2}-4x+16=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
\frac{1}{16}x^{2}-4x+16-16=-16
Subtract 16 from both sides of the equation.
\frac{1}{16}x^{2}-4x=-16
Subtracting 16 from itself leaves 0.
\frac{\frac{1}{16}x^{2}-4x}{\frac{1}{16}}=-\frac{16}{\frac{1}{16}}
Multiply both sides by 16.
x^{2}+\left(-\frac{4}{\frac{1}{16}}\right)x=-\frac{16}{\frac{1}{16}}
Dividing by \frac{1}{16} undoes the multiplication by \frac{1}{16}.
x^{2}-64x=-\frac{16}{\frac{1}{16}}
Divide -4 by \frac{1}{16} by multiplying -4 by the reciprocal of \frac{1}{16}.
x^{2}-64x=-256
Divide -16 by \frac{1}{16} by multiplying -16 by the reciprocal of \frac{1}{16}.
x^{2}-64x+\left(-32\right)^{2}=-256+\left(-32\right)^{2}
Divide -64, the coefficient of the x term, by 2 to get -32. Then add the square of -32 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-64x+1024=-256+1024
Square -32.
x^{2}-64x+1024=768
Add -256 to 1024.
\left(x-32\right)^{2}=768
Factor x^{2}-64x+1024. 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-32\right)^{2}}=\sqrt{768}
Take the square root of both sides of the equation.
x-32=16\sqrt{3} x-32=-16\sqrt{3}
Simplify.
x=16\sqrt{3}+32 x=32-16\sqrt{3}
Add 32 to both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}