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x^{2}-16x+54=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 54}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-16\right)±\sqrt{256-4\times 54}}{2}
Square -16.
x=\frac{-\left(-16\right)±\sqrt{256-216}}{2}
Multiply -4 times 54.
x=\frac{-\left(-16\right)±\sqrt{40}}{2}
Add 256 to -216.
x=\frac{-\left(-16\right)±2\sqrt{10}}{2}
Take the square root of 40.
x=\frac{16±2\sqrt{10}}{2}
The opposite of -16 is 16.
x=\frac{2\sqrt{10}+16}{2}
Now solve the equation x=\frac{16±2\sqrt{10}}{2} when ± is plus. Add 16 to 2\sqrt{10}.
x=\sqrt{10}+8
Divide 16+2\sqrt{10} by 2.
x=\frac{16-2\sqrt{10}}{2}
Now solve the equation x=\frac{16±2\sqrt{10}}{2} when ± is minus. Subtract 2\sqrt{10} from 16.
x=8-\sqrt{10}
Divide 16-2\sqrt{10} by 2.
x=\sqrt{10}+8 x=8-\sqrt{10}
The equation is now solved.
x^{2}-16x+54=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.
x^{2}-16x+54-54=-54
Subtract 54 from both sides of the equation.
x^{2}-16x=-54
Subtracting 54 from itself leaves 0.
x^{2}-16x+\left(-8\right)^{2}=-54+\left(-8\right)^{2}
Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-16x+64=-54+64
Square -8.
x^{2}-16x+64=10
Add -54 to 64.
\left(x-8\right)^{2}=10
Factor x^{2}-16x+64. 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-8\right)^{2}}=\sqrt{10}
Take the square root of both sides of the equation.
x-8=\sqrt{10} x-8=-\sqrt{10}
Simplify.
x=\sqrt{10}+8 x=8-\sqrt{10}
Add 8 to both sides of the equation.
x ^ 2 -16x +54 = 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.
r + s = 16 rs = 54
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 = 8 - u s = 8 + u
Two numbers r and s sum up to 16 exactly when the average of the two numbers is \frac{1}{2}*16 = 8. 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>
(8 - u) (8 + u) = 54
To solve for unknown quantity u, substitute these in the product equation rs = 54
64 - u^2 = 54
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 54-64 = -10
Simplify the expression by subtracting 64 on both sides
u^2 = 10 u = \pm\sqrt{10} = \pm \sqrt{10}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =8 - \sqrt{10} = 4.838 s = 8 + \sqrt{10} = 11.162
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.