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4x^{2}-16x-1=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(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 4\left(-1\right)}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -16 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-16\right)±\sqrt{256-4\times 4\left(-1\right)}}{2\times 4}
Square -16.
x=\frac{-\left(-16\right)±\sqrt{256-16\left(-1\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-\left(-16\right)±\sqrt{256+16}}{2\times 4}
Multiply -16 times -1.
x=\frac{-\left(-16\right)±\sqrt{272}}{2\times 4}
Add 256 to 16.
x=\frac{-\left(-16\right)±4\sqrt{17}}{2\times 4}
Take the square root of 272.
x=\frac{16±4\sqrt{17}}{2\times 4}
The opposite of -16 is 16.
x=\frac{16±4\sqrt{17}}{8}
Multiply 2 times 4.
x=\frac{4\sqrt{17}+16}{8}
Now solve the equation x=\frac{16±4\sqrt{17}}{8} when ± is plus. Add 16 to 4\sqrt{17}.
x=\frac{\sqrt{17}}{2}+2
Divide 16+4\sqrt{17} by 8.
x=\frac{16-4\sqrt{17}}{8}
Now solve the equation x=\frac{16±4\sqrt{17}}{8} when ± is minus. Subtract 4\sqrt{17} from 16.
x=-\frac{\sqrt{17}}{2}+2
Divide 16-4\sqrt{17} by 8.
x=\frac{\sqrt{17}}{2}+2 x=-\frac{\sqrt{17}}{2}+2
The equation is now solved.
4x^{2}-16x-1=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.
4x^{2}-16x-1-\left(-1\right)=-\left(-1\right)
Add 1 to both sides of the equation.
4x^{2}-16x=-\left(-1\right)
Subtracting -1 from itself leaves 0.
4x^{2}-16x=1
Subtract -1 from 0.
\frac{4x^{2}-16x}{4}=\frac{1}{4}
Divide both sides by 4.
x^{2}+\left(-\frac{16}{4}\right)x=\frac{1}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}-4x=\frac{1}{4}
Divide -16 by 4.
x^{2}-4x+\left(-2\right)^{2}=\frac{1}{4}+\left(-2\right)^{2}
Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-4x+4=\frac{1}{4}+4
Square -2.
x^{2}-4x+4=\frac{17}{4}
Add \frac{1}{4} to 4.
\left(x-2\right)^{2}=\frac{17}{4}
Factor x^{2}-4x+4. 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-2\right)^{2}}=\sqrt{\frac{17}{4}}
Take the square root of both sides of the equation.
x-2=\frac{\sqrt{17}}{2} x-2=-\frac{\sqrt{17}}{2}
Simplify.
x=\frac{\sqrt{17}}{2}+2 x=-\frac{\sqrt{17}}{2}+2
Add 2 to both sides of the equation.
x ^ 2 -4x -\frac{1}{4} = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 4
r + s = 4 rs = -\frac{1}{4}
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = 2 - u s = 2 + u
Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 2. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(2 - u) (2 + u) = -\frac{1}{4}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{4}
4 - u^2 = -\frac{1}{4}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{1}{4}-4 = -\frac{17}{4}
Simplify the expression by subtracting 4 on both sides
u^2 = \frac{17}{4} u = \pm\sqrt{\frac{17}{4}} = \pm \frac{\sqrt{17}}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =2 - \frac{\sqrt{17}}{2} = -0.062 s = 2 + \frac{\sqrt{17}}{2} = 4.062
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.