Solve for x (complex solution)
x=\sqrt{10}-2\approx 1.16227766
x=-\left(\sqrt{10}+2\right)\approx -5.16227766
Solve for x
x=\sqrt{10}-2\approx 1.16227766
x=-\sqrt{10}-2\approx -5.16227766
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\frac{1}{2}x^{2}+2x-3=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 \frac{1}{2}\left(-3\right)}}{2\times \frac{1}{2}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 2 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\times \frac{1}{2}\left(-3\right)}}{2\times \frac{1}{2}}
Square 2.
x=\frac{-2±\sqrt{4-2\left(-3\right)}}{2\times \frac{1}{2}}
Multiply -4 times \frac{1}{2}.
x=\frac{-2±\sqrt{4+6}}{2\times \frac{1}{2}}
Multiply -2 times -3.
x=\frac{-2±\sqrt{10}}{2\times \frac{1}{2}}
Add 4 to 6.
x=\frac{-2±\sqrt{10}}{1}
Multiply 2 times \frac{1}{2}.
x=\frac{\sqrt{10}-2}{1}
Now solve the equation x=\frac{-2±\sqrt{10}}{1} when ± is plus. Add -2 to \sqrt{10}.
x=\sqrt{10}-2
Divide -2+\sqrt{10} by 1.
x=\frac{-\sqrt{10}-2}{1}
Now solve the equation x=\frac{-2±\sqrt{10}}{1} when ± is minus. Subtract \sqrt{10} from -2.
x=-\sqrt{10}-2
Divide -2-\sqrt{10} by 1.
x=\sqrt{10}-2 x=-\sqrt{10}-2
The equation is now solved.
\frac{1}{2}x^{2}+2x-3=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.
\frac{1}{2}x^{2}+2x-3-\left(-3\right)=-\left(-3\right)
Add 3 to both sides of the equation.
\frac{1}{2}x^{2}+2x=-\left(-3\right)
Subtracting -3 from itself leaves 0.
\frac{1}{2}x^{2}+2x=3
Subtract -3 from 0.
\frac{\frac{1}{2}x^{2}+2x}{\frac{1}{2}}=\frac{3}{\frac{1}{2}}
Multiply both sides by 2.
x^{2}+\frac{2}{\frac{1}{2}}x=\frac{3}{\frac{1}{2}}
Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.
x^{2}+4x=\frac{3}{\frac{1}{2}}
Divide 2 by \frac{1}{2} by multiplying 2 by the reciprocal of \frac{1}{2}.
x^{2}+4x=6
Divide 3 by \frac{1}{2} by multiplying 3 by the reciprocal of \frac{1}{2}.
x^{2}+4x+2^{2}=6+2^{2}
Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+4x+4=6+4
Square 2.
x^{2}+4x+4=10
Add 6 to 4.
\left(x+2\right)^{2}=10
Factor x^{2}+4x+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+2\right)^{2}}=\sqrt{10}
Take the square root of both sides of the equation.
x+2=\sqrt{10} x+2=-\sqrt{10}
Simplify.
x=\sqrt{10}-2 x=-\sqrt{10}-2
Subtract 2 from both sides of the equation.
\frac{1}{2}x^{2}+2x-3=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 \frac{1}{2}\left(-3\right)}}{2\times \frac{1}{2}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 2 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\times \frac{1}{2}\left(-3\right)}}{2\times \frac{1}{2}}
Square 2.
x=\frac{-2±\sqrt{4-2\left(-3\right)}}{2\times \frac{1}{2}}
Multiply -4 times \frac{1}{2}.
x=\frac{-2±\sqrt{4+6}}{2\times \frac{1}{2}}
Multiply -2 times -3.
x=\frac{-2±\sqrt{10}}{2\times \frac{1}{2}}
Add 4 to 6.
x=\frac{-2±\sqrt{10}}{1}
Multiply 2 times \frac{1}{2}.
x=\frac{\sqrt{10}-2}{1}
Now solve the equation x=\frac{-2±\sqrt{10}}{1} when ± is plus. Add -2 to \sqrt{10}.
x=\sqrt{10}-2
Divide -2+\sqrt{10} by 1.
x=\frac{-\sqrt{10}-2}{1}
Now solve the equation x=\frac{-2±\sqrt{10}}{1} when ± is minus. Subtract \sqrt{10} from -2.
x=-\sqrt{10}-2
Divide -2-\sqrt{10} by 1.
x=\sqrt{10}-2 x=-\sqrt{10}-2
The equation is now solved.
\frac{1}{2}x^{2}+2x-3=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.
\frac{1}{2}x^{2}+2x-3-\left(-3\right)=-\left(-3\right)
Add 3 to both sides of the equation.
\frac{1}{2}x^{2}+2x=-\left(-3\right)
Subtracting -3 from itself leaves 0.
\frac{1}{2}x^{2}+2x=3
Subtract -3 from 0.
\frac{\frac{1}{2}x^{2}+2x}{\frac{1}{2}}=\frac{3}{\frac{1}{2}}
Multiply both sides by 2.
x^{2}+\frac{2}{\frac{1}{2}}x=\frac{3}{\frac{1}{2}}
Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.
x^{2}+4x=\frac{3}{\frac{1}{2}}
Divide 2 by \frac{1}{2} by multiplying 2 by the reciprocal of \frac{1}{2}.
x^{2}+4x=6
Divide 3 by \frac{1}{2} by multiplying 3 by the reciprocal of \frac{1}{2}.
x^{2}+4x+2^{2}=6+2^{2}
Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+4x+4=6+4
Square 2.
x^{2}+4x+4=10
Add 6 to 4.
\left(x+2\right)^{2}=10
Factor x^{2}+4x+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+2\right)^{2}}=\sqrt{10}
Take the square root of both sides of the equation.
x+2=\sqrt{10} x+2=-\sqrt{10}
Simplify.
x=\sqrt{10}-2 x=-\sqrt{10}-2
Subtract 2 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}