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5x^{2}+25x+5=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{-25±\sqrt{25^{2}-4\times 5\times 5}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 25 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-25±\sqrt{625-4\times 5\times 5}}{2\times 5}
Square 25.
x=\frac{-25±\sqrt{625-20\times 5}}{2\times 5}
Multiply -4 times 5.
x=\frac{-25±\sqrt{625-100}}{2\times 5}
Multiply -20 times 5.
x=\frac{-25±\sqrt{525}}{2\times 5}
Add 625 to -100.
x=\frac{-25±5\sqrt{21}}{2\times 5}
Take the square root of 525.
x=\frac{-25±5\sqrt{21}}{10}
Multiply 2 times 5.
x=\frac{5\sqrt{21}-25}{10}
Now solve the equation x=\frac{-25±5\sqrt{21}}{10} when ± is plus. Add -25 to 5\sqrt{21}.
x=\frac{\sqrt{21}-5}{2}
Divide -25+5\sqrt{21} by 10.
x=\frac{-5\sqrt{21}-25}{10}
Now solve the equation x=\frac{-25±5\sqrt{21}}{10} when ± is minus. Subtract 5\sqrt{21} from -25.
x=\frac{-\sqrt{21}-5}{2}
Divide -25-5\sqrt{21} by 10.
x=\frac{\sqrt{21}-5}{2} x=\frac{-\sqrt{21}-5}{2}
The equation is now solved.
5x^{2}+25x+5=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.
5x^{2}+25x+5-5=-5
Subtract 5 from both sides of the equation.
5x^{2}+25x=-5
Subtracting 5 from itself leaves 0.
\frac{5x^{2}+25x}{5}=-\frac{5}{5}
Divide both sides by 5.
x^{2}+\frac{25}{5}x=-\frac{5}{5}
Dividing by 5 undoes the multiplication by 5.
x^{2}+5x=-\frac{5}{5}
Divide 25 by 5.
x^{2}+5x=-1
Divide -5 by 5.
x^{2}+5x+\left(\frac{5}{2}\right)^{2}=-1+\left(\frac{5}{2}\right)^{2}
Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+5x+\frac{25}{4}=-1+\frac{25}{4}
Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+5x+\frac{25}{4}=\frac{21}{4}
Add -1 to \frac{25}{4}.
\left(x+\frac{5}{2}\right)^{2}=\frac{21}{4}
Factor x^{2}+5x+\frac{25}{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{5}{2}\right)^{2}}=\sqrt{\frac{21}{4}}
Take the square root of both sides of the equation.
x+\frac{5}{2}=\frac{\sqrt{21}}{2} x+\frac{5}{2}=-\frac{\sqrt{21}}{2}
Simplify.
x=\frac{\sqrt{21}-5}{2} x=\frac{-\sqrt{21}-5}{2}
Subtract \frac{5}{2} from both sides of the equation.
x ^ 2 +5x +1 = 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 5
r + s = -5 rs = 1
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{5}{2} - u s = -\frac{5}{2} + u
Two numbers r and s sum up to -5 exactly when the average of the two numbers is \frac{1}{2}*-5 = -\frac{5}{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{5}{2} - u) (-\frac{5}{2} + u) = 1
To solve for unknown quantity u, substitute these in the product equation rs = 1
\frac{25}{4} - u^2 = 1
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 1-\frac{25}{4} = -\frac{21}{4}
Simplify the expression by subtracting \frac{25}{4} on both sides
u^2 = \frac{21}{4} u = \pm\sqrt{\frac{21}{4}} = \pm \frac{\sqrt{21}}{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{5}{2} - \frac{\sqrt{21}}{2} = -4.791 s = -\frac{5}{2} + \frac{\sqrt{21}}{2} = -0.209
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.