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x^{2}+7x+11=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{-7±\sqrt{7^{2}-4\times 11}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 7 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-7±\sqrt{49-4\times 11}}{2}
Square 7.
x=\frac{-7±\sqrt{49-44}}{2}
Multiply -4 times 11.
x=\frac{-7±\sqrt{5}}{2}
Add 49 to -44.
x=\frac{\sqrt{5}-7}{2}
Now solve the equation x=\frac{-7±\sqrt{5}}{2} when ± is plus. Add -7 to \sqrt{5}.
x=\frac{-\sqrt{5}-7}{2}
Now solve the equation x=\frac{-7±\sqrt{5}}{2} when ± is minus. Subtract \sqrt{5} from -7.
x=\frac{\sqrt{5}-7}{2} x=\frac{-\sqrt{5}-7}{2}
The equation is now solved.
x^{2}+7x+11=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}+7x+11-11=-11
Subtract 11 from both sides of the equation.
x^{2}+7x=-11
Subtracting 11 from itself leaves 0.
x^{2}+7x+\left(\frac{7}{2}\right)^{2}=-11+\left(\frac{7}{2}\right)^{2}
Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+7x+\frac{49}{4}=-11+\frac{49}{4}
Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+7x+\frac{49}{4}=\frac{5}{4}
Add -11 to \frac{49}{4}.
\left(x+\frac{7}{2}\right)^{2}=\frac{5}{4}
Factor x^{2}+7x+\frac{49}{4}. 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{7}{2}\right)^{2}}=\sqrt{\frac{5}{4}}
Take the square root of both sides of the equation.
x+\frac{7}{2}=\frac{\sqrt{5}}{2} x+\frac{7}{2}=-\frac{\sqrt{5}}{2}
Simplify.
x=\frac{\sqrt{5}-7}{2} x=\frac{-\sqrt{5}-7}{2}
Subtract \frac{7}{2} from both sides of the equation.
x ^ 2 +7x +11 = 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 = -7 rs = 11
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{7}{2} - u s = -\frac{7}{2} + u
Two numbers r and s sum up to -7 exactly when the average of the two numbers is \frac{1}{2}*-7 = -\frac{7}{2}. 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{7}{2} - u) (-\frac{7}{2} + u) = 11
To solve for unknown quantity u, substitute these in the product equation rs = 11
\frac{49}{4} - u^2 = 11
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 11-\frac{49}{4} = -\frac{5}{4}
Simplify the expression by subtracting \frac{49}{4} on both sides
u^2 = \frac{5}{4} u = \pm\sqrt{\frac{5}{4}} = \pm \frac{\sqrt{5}}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{7}{2} - \frac{\sqrt{5}}{2} = -4.618 s = -\frac{7}{2} + \frac{\sqrt{5}}{2} = -2.382
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.