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7x^{2}+19x-4=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{-19±\sqrt{19^{2}-4\times 7\left(-4\right)}}{2\times 7}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 19 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-19±\sqrt{361-4\times 7\left(-4\right)}}{2\times 7}
Square 19.
x=\frac{-19±\sqrt{361-28\left(-4\right)}}{2\times 7}
Multiply -4 times 7.
x=\frac{-19±\sqrt{361+112}}{2\times 7}
Multiply -28 times -4.
x=\frac{-19±\sqrt{473}}{2\times 7}
Add 361 to 112.
x=\frac{-19±\sqrt{473}}{14}
Multiply 2 times 7.
x=\frac{\sqrt{473}-19}{14}
Now solve the equation x=\frac{-19±\sqrt{473}}{14} when ± is plus. Add -19 to \sqrt{473}.
x=\frac{-\sqrt{473}-19}{14}
Now solve the equation x=\frac{-19±\sqrt{473}}{14} when ± is minus. Subtract \sqrt{473} from -19.
x=\frac{\sqrt{473}-19}{14} x=\frac{-\sqrt{473}-19}{14}
The equation is now solved.
7x^{2}+19x-4=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.
7x^{2}+19x-4-\left(-4\right)=-\left(-4\right)
Add 4 to both sides of the equation.
7x^{2}+19x=-\left(-4\right)
Subtracting -4 from itself leaves 0.
7x^{2}+19x=4
Subtract -4 from 0.
\frac{7x^{2}+19x}{7}=\frac{4}{7}
Divide both sides by 7.
x^{2}+\frac{19}{7}x=\frac{4}{7}
Dividing by 7 undoes the multiplication by 7.
x^{2}+\frac{19}{7}x+\left(\frac{19}{14}\right)^{2}=\frac{4}{7}+\left(\frac{19}{14}\right)^{2}
Divide \frac{19}{7}, the coefficient of the x term, by 2 to get \frac{19}{14}. Then add the square of \frac{19}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{19}{7}x+\frac{361}{196}=\frac{4}{7}+\frac{361}{196}
Square \frac{19}{14} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{19}{7}x+\frac{361}{196}=\frac{473}{196}
Add \frac{4}{7} to \frac{361}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{19}{14}\right)^{2}=\frac{473}{196}
Factor x^{2}+\frac{19}{7}x+\frac{361}{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(x+\frac{19}{14}\right)^{2}}=\sqrt{\frac{473}{196}}
Take the square root of both sides of the equation.
x+\frac{19}{14}=\frac{\sqrt{473}}{14} x+\frac{19}{14}=-\frac{\sqrt{473}}{14}
Simplify.
x=\frac{\sqrt{473}-19}{14} x=\frac{-\sqrt{473}-19}{14}
Subtract \frac{19}{14} from both sides of the equation.
x ^ 2 +\frac{19}{7}x -\frac{4}{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{19}{7} rs = -\frac{4}{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{19}{14} - u s = -\frac{19}{14} + u
Two numbers r and s sum up to -\frac{19}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{19}{7} = -\frac{19}{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{19}{14} - u) (-\frac{19}{14} + u) = -\frac{4}{7}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{7}
\frac{361}{196} - u^2 = -\frac{4}{7}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{4}{7}-\frac{361}{196} = -\frac{473}{196}
Simplify the expression by subtracting \frac{361}{196} on both sides
u^2 = \frac{473}{196} u = \pm\sqrt{\frac{473}{196}} = \pm \frac{\sqrt{473}}{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{19}{14} - \frac{\sqrt{473}}{14} = -2.911 s = -\frac{19}{14} + \frac{\sqrt{473}}{14} = 0.196
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.