Solve for x (complex solution)
x=\sqrt{91}-10\approx -0.460607986
x=-\left(\sqrt{91}+10\right)\approx -19.539392014
Solve for x
x=\sqrt{91}-10\approx -0.460607986
x=-\sqrt{91}-10\approx -19.539392014
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x^{2}+20x+9=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{-20±\sqrt{20^{2}-4\times 9}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-20±\sqrt{400-4\times 9}}{2}
Square 20.
x=\frac{-20±\sqrt{400-36}}{2}
Multiply -4 times 9.
x=\frac{-20±\sqrt{364}}{2}
Add 400 to -36.
x=\frac{-20±2\sqrt{91}}{2}
Take the square root of 364.
x=\frac{2\sqrt{91}-20}{2}
Now solve the equation x=\frac{-20±2\sqrt{91}}{2} when ± is plus. Add -20 to 2\sqrt{91}.
x=\sqrt{91}-10
Divide -20+2\sqrt{91} by 2.
x=\frac{-2\sqrt{91}-20}{2}
Now solve the equation x=\frac{-20±2\sqrt{91}}{2} when ± is minus. Subtract 2\sqrt{91} from -20.
x=-\sqrt{91}-10
Divide -20-2\sqrt{91} by 2.
x=\sqrt{91}-10 x=-\sqrt{91}-10
The equation is now solved.
x^{2}+20x+9=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}+20x+9-9=-9
Subtract 9 from both sides of the equation.
x^{2}+20x=-9
Subtracting 9 from itself leaves 0.
x^{2}+20x+10^{2}=-9+10^{2}
Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+20x+100=-9+100
Square 10.
x^{2}+20x+100=91
Add -9 to 100.
\left(x+10\right)^{2}=91
Factor x^{2}+20x+100. 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+10\right)^{2}}=\sqrt{91}
Take the square root of both sides of the equation.
x+10=\sqrt{91} x+10=-\sqrt{91}
Simplify.
x=\sqrt{91}-10 x=-\sqrt{91}-10
Subtract 10 from both sides of the equation.
x ^ 2 +20x +9 = 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 = -20 rs = 9
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 = -10 - u s = -10 + u
Two numbers r and s sum up to -20 exactly when the average of the two numbers is \frac{1}{2}*-20 = -10. 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>
(-10 - u) (-10 + u) = 9
To solve for unknown quantity u, substitute these in the product equation rs = 9
100 - u^2 = 9
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 9-100 = -91
Simplify the expression by subtracting 100 on both sides
u^2 = 91 u = \pm\sqrt{91} = \pm \sqrt{91}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-10 - \sqrt{91} = -19.539 s = -10 + \sqrt{91} = -0.461
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
x^{2}+20x+9=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{-20±\sqrt{20^{2}-4\times 9}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-20±\sqrt{400-4\times 9}}{2}
Square 20.
x=\frac{-20±\sqrt{400-36}}{2}
Multiply -4 times 9.
x=\frac{-20±\sqrt{364}}{2}
Add 400 to -36.
x=\frac{-20±2\sqrt{91}}{2}
Take the square root of 364.
x=\frac{2\sqrt{91}-20}{2}
Now solve the equation x=\frac{-20±2\sqrt{91}}{2} when ± is plus. Add -20 to 2\sqrt{91}.
x=\sqrt{91}-10
Divide -20+2\sqrt{91} by 2.
x=\frac{-2\sqrt{91}-20}{2}
Now solve the equation x=\frac{-20±2\sqrt{91}}{2} when ± is minus. Subtract 2\sqrt{91} from -20.
x=-\sqrt{91}-10
Divide -20-2\sqrt{91} by 2.
x=\sqrt{91}-10 x=-\sqrt{91}-10
The equation is now solved.
x^{2}+20x+9=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}+20x+9-9=-9
Subtract 9 from both sides of the equation.
x^{2}+20x=-9
Subtracting 9 from itself leaves 0.
x^{2}+20x+10^{2}=-9+10^{2}
Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+20x+100=-9+100
Square 10.
x^{2}+20x+100=91
Add -9 to 100.
\left(x+10\right)^{2}=91
Factor x^{2}+20x+100. 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+10\right)^{2}}=\sqrt{91}
Take the square root of both sides of the equation.
x+10=\sqrt{91} x+10=-\sqrt{91}
Simplify.
x=\sqrt{91}-10 x=-\sqrt{91}-10
Subtract 10 from both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}