Solve for x
x=\frac{6}{7}\approx 0.857142857
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49x^{2}-84x+36=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(-84\right)±\sqrt{\left(-84\right)^{2}-4\times 49\times 36}}{2\times 49}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 49 for a, -84 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-84\right)±\sqrt{7056-4\times 49\times 36}}{2\times 49}
Square -84.
x=\frac{-\left(-84\right)±\sqrt{7056-196\times 36}}{2\times 49}
Multiply -4 times 49.
x=\frac{-\left(-84\right)±\sqrt{7056-7056}}{2\times 49}
Multiply -196 times 36.
x=\frac{-\left(-84\right)±\sqrt{0}}{2\times 49}
Add 7056 to -7056.
x=-\frac{-84}{2\times 49}
Take the square root of 0.
x=\frac{84}{2\times 49}
The opposite of -84 is 84.
x=\frac{84}{98}
Multiply 2 times 49.
x=\frac{6}{7}
Reduce the fraction \frac{84}{98} to lowest terms by extracting and canceling out 14.
49x^{2}-84x+36=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.
49x^{2}-84x+36-36=-36
Subtract 36 from both sides of the equation.
49x^{2}-84x=-36
Subtracting 36 from itself leaves 0.
\frac{49x^{2}-84x}{49}=-\frac{36}{49}
Divide both sides by 49.
x^{2}+\left(-\frac{84}{49}\right)x=-\frac{36}{49}
Dividing by 49 undoes the multiplication by 49.
x^{2}-\frac{12}{7}x=-\frac{36}{49}
Reduce the fraction \frac{-84}{49} to lowest terms by extracting and canceling out 7.
x^{2}-\frac{12}{7}x+\left(-\frac{6}{7}\right)^{2}=-\frac{36}{49}+\left(-\frac{6}{7}\right)^{2}
Divide -\frac{12}{7}, the coefficient of the x term, by 2 to get -\frac{6}{7}. Then add the square of -\frac{6}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{12}{7}x+\frac{36}{49}=\frac{-36+36}{49}
Square -\frac{6}{7} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{12}{7}x+\frac{36}{49}=0
Add -\frac{36}{49} to \frac{36}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{6}{7}\right)^{2}=0
Factor x^{2}-\frac{12}{7}x+\frac{36}{49}. 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-\frac{6}{7}\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
x-\frac{6}{7}=0 x-\frac{6}{7}=0
Simplify.
x=\frac{6}{7} x=\frac{6}{7}
Add \frac{6}{7} to both sides of the equation.
x=\frac{6}{7}
The equation is now solved. Solutions are the same.
x ^ 2 -\frac{12}{7}x +\frac{36}{49} = 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 49
r + s = \frac{12}{7} rs = \frac{36}{49}
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 = \frac{6}{7} - u s = \frac{6}{7} + u
Two numbers r and s sum up to \frac{12}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{12}{7} = \frac{6}{7}. 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>
(\frac{6}{7} - u) (\frac{6}{7} + u) = \frac{36}{49}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{36}{49}
\frac{36}{49} - u^2 = \frac{36}{49}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{36}{49}-\frac{36}{49} = 0
Simplify the expression by subtracting \frac{36}{49} on both sides
u^2 = 0 u = 0
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{6}{7} - 0 s = \frac{6}{7} + 0
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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