Solve for x
x=\sqrt{94}+38\approx 47.695359715
x=38-\sqrt{94}\approx 28.304640285
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x^{2}-76x+1350=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(-76\right)±\sqrt{\left(-76\right)^{2}-4\times 1350}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -76 for b, and 1350 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-76\right)±\sqrt{5776-4\times 1350}}{2}
Square -76.
x=\frac{-\left(-76\right)±\sqrt{5776-5400}}{2}
Multiply -4 times 1350.
x=\frac{-\left(-76\right)±\sqrt{376}}{2}
Add 5776 to -5400.
x=\frac{-\left(-76\right)±2\sqrt{94}}{2}
Take the square root of 376.
x=\frac{76±2\sqrt{94}}{2}
The opposite of -76 is 76.
x=\frac{2\sqrt{94}+76}{2}
Now solve the equation x=\frac{76±2\sqrt{94}}{2} when ± is plus. Add 76 to 2\sqrt{94}.
x=\sqrt{94}+38
Divide 76+2\sqrt{94} by 2.
x=\frac{76-2\sqrt{94}}{2}
Now solve the equation x=\frac{76±2\sqrt{94}}{2} when ± is minus. Subtract 2\sqrt{94} from 76.
x=38-\sqrt{94}
Divide 76-2\sqrt{94} by 2.
x=\sqrt{94}+38 x=38-\sqrt{94}
The equation is now solved.
x^{2}-76x+1350=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}-76x+1350-1350=-1350
Subtract 1350 from both sides of the equation.
x^{2}-76x=-1350
Subtracting 1350 from itself leaves 0.
x^{2}-76x+\left(-38\right)^{2}=-1350+\left(-38\right)^{2}
Divide -76, the coefficient of the x term, by 2 to get -38. Then add the square of -38 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-76x+1444=-1350+1444
Square -38.
x^{2}-76x+1444=94
Add -1350 to 1444.
\left(x-38\right)^{2}=94
Factor x^{2}-76x+1444. 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-38\right)^{2}}=\sqrt{94}
Take the square root of both sides of the equation.
x-38=\sqrt{94} x-38=-\sqrt{94}
Simplify.
x=\sqrt{94}+38 x=38-\sqrt{94}
Add 38 to both sides of the equation.
x ^ 2 -76x +1350 = 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 = 76 rs = 1350
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 = 38 - u s = 38 + u
Two numbers r and s sum up to 76 exactly when the average of the two numbers is \frac{1}{2}*76 = 38. 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>
(38 - u) (38 + u) = 1350
To solve for unknown quantity u, substitute these in the product equation rs = 1350
1444 - u^2 = 1350
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 1350-1444 = -94
Simplify the expression by subtracting 1444 on both sides
u^2 = 94 u = \pm\sqrt{94} = \pm \sqrt{94}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =38 - \sqrt{94} = 28.305 s = 38 + \sqrt{94} = 47.695
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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