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3x^{2}+6x-28=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(-28\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 6 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-6±\sqrt{36-4\times 3\left(-28\right)}}{2\times 3}
Square 6.
x=\frac{-6±\sqrt{36-12\left(-28\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-6±\sqrt{36+336}}{2\times 3}
Multiply -12 times -28.
x=\frac{-6±\sqrt{372}}{2\times 3}
Add 36 to 336.
x=\frac{-6±2\sqrt{93}}{2\times 3}
Take the square root of 372.
x=\frac{-6±2\sqrt{93}}{6}
Multiply 2 times 3.
x=\frac{2\sqrt{93}-6}{6}
Now solve the equation x=\frac{-6±2\sqrt{93}}{6} when ± is plus. Add -6 to 2\sqrt{93}.
x=\frac{\sqrt{93}}{3}-1
Divide -6+2\sqrt{93} by 6.
x=\frac{-2\sqrt{93}-6}{6}
Now solve the equation x=\frac{-6±2\sqrt{93}}{6} when ± is minus. Subtract 2\sqrt{93} from -6.
x=-\frac{\sqrt{93}}{3}-1
Divide -6-2\sqrt{93} by 6.
x=\frac{\sqrt{93}}{3}-1 x=-\frac{\sqrt{93}}{3}-1
The equation is now solved.
3x^{2}+6x-28=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-28-\left(-28\right)=-\left(-28\right)
Add 28 to both sides of the equation.
3x^{2}+6x=-\left(-28\right)
Subtracting -28 from itself leaves 0.
3x^{2}+6x=28
Subtract -28 from 0.
\frac{3x^{2}+6x}{3}=\frac{28}{3}
Divide both sides by 3.
x^{2}+\frac{6}{3}x=\frac{28}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+2x=\frac{28}{3}
Divide 6 by 3.
x^{2}+2x+1^{2}=\frac{28}{3}+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=\frac{28}{3}+1
Square 1.
x^{2}+2x+1=\frac{31}{3}
Add \frac{28}{3} to 1.
\left(x+1\right)^{2}=\frac{31}{3}
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{\frac{31}{3}}
Take the square root of both sides of the equation.
x+1=\frac{\sqrt{93}}{3} x+1=-\frac{\sqrt{93}}{3}
Simplify.
x=\frac{\sqrt{93}}{3}-1 x=-\frac{\sqrt{93}}{3}-1
Subtract 1 from both sides of the equation.