Solve for x
x=\frac{\sqrt{21}-3}{2}\approx 0.791287847
x=\frac{-\sqrt{21}-3}{2}\approx -3.791287847
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x^{2}+3x-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{-3±\sqrt{3^{2}-4\left(-3\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-3±\sqrt{9-4\left(-3\right)}}{2}
Square 3.
x=\frac{-3±\sqrt{9+12}}{2}
Multiply -4 times -3.
x=\frac{-3±\sqrt{21}}{2}
Add 9 to 12.
x=\frac{\sqrt{21}-3}{2}
Now solve the equation x=\frac{-3±\sqrt{21}}{2} when ± is plus. Add -3 to \sqrt{21}.
x=\frac{-\sqrt{21}-3}{2}
Now solve the equation x=\frac{-3±\sqrt{21}}{2} when ± is minus. Subtract \sqrt{21} from -3.
x=\frac{\sqrt{21}-3}{2} x=\frac{-\sqrt{21}-3}{2}
The equation is now solved.
x^{2}+3x-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.
x^{2}+3x-3-\left(-3\right)=-\left(-3\right)
Add 3 to both sides of the equation.
x^{2}+3x=-\left(-3\right)
Subtracting -3 from itself leaves 0.
x^{2}+3x=3
Subtract -3 from 0.
x^{2}+3x+\left(\frac{3}{2}\right)^{2}=3+\left(\frac{3}{2}\right)^{2}
Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+3x+\frac{9}{4}=3+\frac{9}{4}
Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+3x+\frac{9}{4}=\frac{21}{4}
Add 3 to \frac{9}{4}.
\left(x+\frac{3}{2}\right)^{2}=\frac{21}{4}
Factor x^{2}+3x+\frac{9}{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{3}{2}\right)^{2}}=\sqrt{\frac{21}{4}}
Take the square root of both sides of the equation.
x+\frac{3}{2}=\frac{\sqrt{21}}{2} x+\frac{3}{2}=-\frac{\sqrt{21}}{2}
Simplify.
x=\frac{\sqrt{21}-3}{2} x=\frac{-\sqrt{21}-3}{2}
Subtract \frac{3}{2} from both sides of the equation.
x ^ 2 +3x -3 = 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 = -3 rs = -3
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{3}{2} - u s = -\frac{3}{2} + u
Two numbers r and s sum up to -3 exactly when the average of the two numbers is \frac{1}{2}*-3 = -\frac{3}{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{3}{2} - u) (-\frac{3}{2} + u) = -3
To solve for unknown quantity u, substitute these in the product equation rs = -3
\frac{9}{4} - u^2 = -3
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -3-\frac{9}{4} = -\frac{21}{4}
Simplify the expression by subtracting \frac{9}{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{3}{2} - \frac{\sqrt{21}}{2} = -3.791 s = -\frac{3}{2} + \frac{\sqrt{21}}{2} = 0.791
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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}