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7x^{2}+2x+8=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{-2±\sqrt{2^{2}-4\times 7\times 8}}{2\times 7}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 2 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\times 7\times 8}}{2\times 7}
Square 2.
x=\frac{-2±\sqrt{4-28\times 8}}{2\times 7}
Multiply -4 times 7.
x=\frac{-2±\sqrt{4-224}}{2\times 7}
Multiply -28 times 8.
x=\frac{-2±\sqrt{-220}}{2\times 7}
Add 4 to -224.
x=\frac{-2±2\sqrt{55}i}{2\times 7}
Take the square root of -220.
x=\frac{-2±2\sqrt{55}i}{14}
Multiply 2 times 7.
x=\frac{-2+2\sqrt{55}i}{14}
Now solve the equation x=\frac{-2±2\sqrt{55}i}{14} when ± is plus. Add -2 to 2i\sqrt{55}.
x=\frac{-1+\sqrt{55}i}{7}
Divide -2+2i\sqrt{55} by 14.
x=\frac{-2\sqrt{55}i-2}{14}
Now solve the equation x=\frac{-2±2\sqrt{55}i}{14} when ± is minus. Subtract 2i\sqrt{55} from -2.
x=\frac{-\sqrt{55}i-1}{7}
Divide -2-2i\sqrt{55} by 14.
x=\frac{-1+\sqrt{55}i}{7} x=\frac{-\sqrt{55}i-1}{7}
The equation is now solved.
7x^{2}+2x+8=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}+2x+8-8=-8
Subtract 8 from both sides of the equation.
7x^{2}+2x=-8
Subtracting 8 from itself leaves 0.
\frac{7x^{2}+2x}{7}=-\frac{8}{7}
Divide both sides by 7.
x^{2}+\frac{2}{7}x=-\frac{8}{7}
Dividing by 7 undoes the multiplication by 7.
x^{2}+\frac{2}{7}x+\left(\frac{1}{7}\right)^{2}=-\frac{8}{7}+\left(\frac{1}{7}\right)^{2}
Divide \frac{2}{7}, the coefficient of the x term, by 2 to get \frac{1}{7}. Then add the square of \frac{1}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{2}{7}x+\frac{1}{49}=-\frac{8}{7}+\frac{1}{49}
Square \frac{1}{7} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{2}{7}x+\frac{1}{49}=-\frac{55}{49}
Add -\frac{8}{7} to \frac{1}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{1}{7}\right)^{2}=-\frac{55}{49}
Factor x^{2}+\frac{2}{7}x+\frac{1}{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{1}{7}\right)^{2}}=\sqrt{-\frac{55}{49}}
Take the square root of both sides of the equation.
x+\frac{1}{7}=\frac{\sqrt{55}i}{7} x+\frac{1}{7}=-\frac{\sqrt{55}i}{7}
Simplify.
x=\frac{-1+\sqrt{55}i}{7} x=\frac{-\sqrt{55}i-1}{7}
Subtract \frac{1}{7} from both sides of the equation.
x ^ 2 +\frac{2}{7}x +\frac{8}{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{2}{7} rs = \frac{8}{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{1}{7} - u s = -\frac{1}{7} + u
Two numbers r and s sum up to -\frac{2}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{2}{7} = -\frac{1}{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{1}{7} - u) (-\frac{1}{7} + u) = \frac{8}{7}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{8}{7}
\frac{1}{49} - u^2 = \frac{8}{7}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{8}{7}-\frac{1}{49} = \frac{55}{49}
Simplify the expression by subtracting \frac{1}{49} on both sides
u^2 = -\frac{55}{49} u = \pm\sqrt{-\frac{55}{49}} = \pm \frac{\sqrt{55}}{7}i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{1}{7} - \frac{\sqrt{55}}{7}i = -0.143 - 1.059i s = -\frac{1}{7} + \frac{\sqrt{55}}{7}i = -0.143 + 1.059i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.