Solve for x
x=3\sqrt{3}-5\approx 0.196152423
x=-3\sqrt{3}-5\approx -10.196152423
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x^{2}+10x=2
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^{2}+10x-2=2-2
Subtract 2 from both sides of the equation.
x^{2}+10x-2=0
Subtracting 2 from itself leaves 0.
x=\frac{-10±\sqrt{10^{2}-4\left(-2\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-10±\sqrt{100-4\left(-2\right)}}{2}
Square 10.
x=\frac{-10±\sqrt{100+8}}{2}
Multiply -4 times -2.
x=\frac{-10±\sqrt{108}}{2}
Add 100 to 8.
x=\frac{-10±6\sqrt{3}}{2}
Take the square root of 108.
x=\frac{6\sqrt{3}-10}{2}
Now solve the equation x=\frac{-10±6\sqrt{3}}{2} when ± is plus. Add -10 to 6\sqrt{3}.
x=3\sqrt{3}-5
Divide -10+6\sqrt{3} by 2.
x=\frac{-6\sqrt{3}-10}{2}
Now solve the equation x=\frac{-10±6\sqrt{3}}{2} when ± is minus. Subtract 6\sqrt{3} from -10.
x=-3\sqrt{3}-5
Divide -10-6\sqrt{3} by 2.
x=3\sqrt{3}-5 x=-3\sqrt{3}-5
The equation is now solved.
x^{2}+10x=2
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}+10x+5^{2}=2+5^{2}
Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+10x+25=2+25
Square 5.
x^{2}+10x+25=27
Add 2 to 25.
\left(x+5\right)^{2}=27
Factor x^{2}+10x+25. 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+5\right)^{2}}=\sqrt{27}
Take the square root of both sides of the equation.
x+5=3\sqrt{3} x+5=-3\sqrt{3}
Simplify.
x=3\sqrt{3}-5 x=-3\sqrt{3}-5
Subtract 5 from both sides of the equation.
Examples
Quadratic equation
{ 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}