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3x^{2}+6x-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{-6±\sqrt{6^{2}-4\times 3\left(-3\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 6 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-6±\sqrt{36-4\times 3\left(-3\right)}}{2\times 3}
Square 6.
x=\frac{-6±\sqrt{36-12\left(-3\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-6±\sqrt{36+36}}{2\times 3}
Multiply -12 times -3.
x=\frac{-6±\sqrt{72}}{2\times 3}
Add 36 to 36.
x=\frac{-6±6\sqrt{2}}{2\times 3}
Take the square root of 72.
x=\frac{-6±6\sqrt{2}}{6}
Multiply 2 times 3.
x=\frac{6\sqrt{2}-6}{6}
Now solve the equation x=\frac{-6±6\sqrt{2}}{6} when ± is plus. Add -6 to 6\sqrt{2}.
x=\sqrt{2}-1
Divide -6+6\sqrt{2} by 6.
x=\frac{-6\sqrt{2}-6}{6}
Now solve the equation x=\frac{-6±6\sqrt{2}}{6} when ± is minus. Subtract 6\sqrt{2} from -6.
x=-\sqrt{2}-1
Divide -6-6\sqrt{2} by 6.
x=\sqrt{2}-1 x=-\sqrt{2}-1
The equation is now solved.
3x^{2}+6x-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.
3x^{2}+6x-3-\left(-3\right)=-\left(-3\right)
Add 3 to both sides of the equation.
3x^{2}+6x=-\left(-3\right)
Subtracting -3 from itself leaves 0.
3x^{2}+6x=3
Subtract -3 from 0.
\frac{3x^{2}+6x}{3}=\frac{3}{3}
Divide both sides by 3.
x^{2}+\frac{6}{3}x=\frac{3}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+2x=\frac{3}{3}
Divide 6 by 3.
x^{2}+2x=1
Divide 3 by 3.
x^{2}+2x+1^{2}=1+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+2x+1=1+1
Square 1.
x^{2}+2x+1=2
Add 1 to 1.
\left(x+1\right)^{2}=2
Factor x^{2}+2x+1. 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+1\right)^{2}}=\sqrt{2}
Take the square root of both sides of the equation.
x+1=\sqrt{2} x+1=-\sqrt{2}
Simplify.
x=\sqrt{2}-1 x=-\sqrt{2}-1
Subtract 1 from both sides of the equation.
x ^ 2 +2x -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 3
r + s = -2 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 = -1 - u s = -1 + u
Two numbers r and s sum up to -2 exactly when the average of the two numbers is \frac{1}{2}*-2 = -1. 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>
(-1 - u) (-1 + u) = -1
To solve for unknown quantity u, substitute these in the product equation rs = -1
1 - u^2 = -1
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -1-1 = -2
Simplify the expression by subtracting 1 on both sides
u^2 = 2 u = \pm\sqrt{2} = \pm \sqrt{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-1 - \sqrt{2} = -2.414 s = -1 + \sqrt{2} = 0.414
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
3x^{2}+6x-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{-6±\sqrt{6^{2}-4\times 3\left(-3\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 6 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-6±\sqrt{36-4\times 3\left(-3\right)}}{2\times 3}
Square 6.
x=\frac{-6±\sqrt{36-12\left(-3\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-6±\sqrt{36+36}}{2\times 3}
Multiply -12 times -3.
x=\frac{-6±\sqrt{72}}{2\times 3}
Add 36 to 36.
x=\frac{-6±6\sqrt{2}}{2\times 3}
Take the square root of 72.
x=\frac{-6±6\sqrt{2}}{6}
Multiply 2 times 3.
x=\frac{6\sqrt{2}-6}{6}
Now solve the equation x=\frac{-6±6\sqrt{2}}{6} when ± is plus. Add -6 to 6\sqrt{2}.
x=\sqrt{2}-1
Divide -6+6\sqrt{2} by 6.
x=\frac{-6\sqrt{2}-6}{6}
Now solve the equation x=\frac{-6±6\sqrt{2}}{6} when ± is minus. Subtract 6\sqrt{2} from -6.
x=-\sqrt{2}-1
Divide -6-6\sqrt{2} by 6.
x=\sqrt{2}-1 x=-\sqrt{2}-1
The equation is now solved.
3x^{2}+6x-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.
3x^{2}+6x-3-\left(-3\right)=-\left(-3\right)
Add 3 to both sides of the equation.
3x^{2}+6x=-\left(-3\right)
Subtracting -3 from itself leaves 0.
3x^{2}+6x=3
Subtract -3 from 0.
\frac{3x^{2}+6x}{3}=\frac{3}{3}
Divide both sides by 3.
x^{2}+\frac{6}{3}x=\frac{3}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+2x=\frac{3}{3}
Divide 6 by 3.
x^{2}+2x=1
Divide 3 by 3.
x^{2}+2x+1^{2}=1+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+2x+1=1+1
Square 1.
x^{2}+2x+1=2
Add 1 to 1.
\left(x+1\right)^{2}=2
Factor x^{2}+2x+1. 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+1\right)^{2}}=\sqrt{2}
Take the square root of both sides of the equation.
x+1=\sqrt{2} x+1=-\sqrt{2}
Simplify.
x=\sqrt{2}-1 x=-\sqrt{2}-1
Subtract 1 from both sides of the equation.