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7m^{2}+9m+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.
m=\frac{-9±\sqrt{9^{2}-4\times 7}}{2\times 7}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 9 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
m=\frac{-9±\sqrt{81-4\times 7}}{2\times 7}
Square 9.
m=\frac{-9±\sqrt{81-28}}{2\times 7}
Multiply -4 times 7.
m=\frac{-9±\sqrt{53}}{2\times 7}
Add 81 to -28.
m=\frac{-9±\sqrt{53}}{14}
Multiply 2 times 7.
m=\frac{\sqrt{53}-9}{14}
Now solve the equation m=\frac{-9±\sqrt{53}}{14} when ± is plus. Add -9 to \sqrt{53}.
m=\frac{-\sqrt{53}-9}{14}
Now solve the equation m=\frac{-9±\sqrt{53}}{14} when ± is minus. Subtract \sqrt{53} from -9.
m=\frac{\sqrt{53}-9}{14} m=\frac{-\sqrt{53}-9}{14}
The equation is now solved.
7m^{2}+9m+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.
7m^{2}+9m+1-1=-1
Subtract 1 from both sides of the equation.
7m^{2}+9m=-1
Subtracting 1 from itself leaves 0.
\frac{7m^{2}+9m}{7}=-\frac{1}{7}
Divide both sides by 7.
m^{2}+\frac{9}{7}m=-\frac{1}{7}
Dividing by 7 undoes the multiplication by 7.
m^{2}+\frac{9}{7}m+\left(\frac{9}{14}\right)^{2}=-\frac{1}{7}+\left(\frac{9}{14}\right)^{2}
Divide \frac{9}{7}, the coefficient of the x term, by 2 to get \frac{9}{14}. Then add the square of \frac{9}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
m^{2}+\frac{9}{7}m+\frac{81}{196}=-\frac{1}{7}+\frac{81}{196}
Square \frac{9}{14} by squaring both the numerator and the denominator of the fraction.
m^{2}+\frac{9}{7}m+\frac{81}{196}=\frac{53}{196}
Add -\frac{1}{7} to \frac{81}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(m+\frac{9}{14}\right)^{2}=\frac{53}{196}
Factor m^{2}+\frac{9}{7}m+\frac{81}{196}. 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(m+\frac{9}{14}\right)^{2}}=\sqrt{\frac{53}{196}}
Take the square root of both sides of the equation.
m+\frac{9}{14}=\frac{\sqrt{53}}{14} m+\frac{9}{14}=-\frac{\sqrt{53}}{14}
Simplify.
m=\frac{\sqrt{53}-9}{14} m=\frac{-\sqrt{53}-9}{14}
Subtract \frac{9}{14} from both sides of the equation.
x ^ 2 +\frac{9}{7}x +\frac{1}{7} = 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 7
r + s = -\frac{9}{7} rs = \frac{1}{7}
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{9}{14} - u s = -\frac{9}{14} + u
Two numbers r and s sum up to -\frac{9}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{9}{7} = -\frac{9}{14}. 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{9}{14} - u) (-\frac{9}{14} + u) = \frac{1}{7}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{7}
\frac{81}{196} - u^2 = \frac{1}{7}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{1}{7}-\frac{81}{196} = -\frac{53}{196}
Simplify the expression by subtracting \frac{81}{196} on both sides
u^2 = \frac{53}{196} u = \pm\sqrt{\frac{53}{196}} = \pm \frac{\sqrt{53}}{14}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{9}{14} - \frac{\sqrt{53}}{14} = -1.163 s = -\frac{9}{14} + \frac{\sqrt{53}}{14} = -0.123
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.