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3x^{2}+22x+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{-22±\sqrt{22^{2}-4\times 3\times 28}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 22 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-22±\sqrt{484-4\times 3\times 28}}{2\times 3}
Square 22.
x=\frac{-22±\sqrt{484-12\times 28}}{2\times 3}
Multiply -4 times 3.
x=\frac{-22±\sqrt{484-336}}{2\times 3}
Multiply -12 times 28.
x=\frac{-22±\sqrt{148}}{2\times 3}
Add 484 to -336.
x=\frac{-22±2\sqrt{37}}{2\times 3}
Take the square root of 148.
x=\frac{-22±2\sqrt{37}}{6}
Multiply 2 times 3.
x=\frac{2\sqrt{37}-22}{6}
Now solve the equation x=\frac{-22±2\sqrt{37}}{6} when ± is plus. Add -22 to 2\sqrt{37}.
x=\frac{\sqrt{37}-11}{3}
Divide -22+2\sqrt{37} by 6.
x=\frac{-2\sqrt{37}-22}{6}
Now solve the equation x=\frac{-22±2\sqrt{37}}{6} when ± is minus. Subtract 2\sqrt{37} from -22.
x=\frac{-\sqrt{37}-11}{3}
Divide -22-2\sqrt{37} by 6.
x=\frac{\sqrt{37}-11}{3} x=\frac{-\sqrt{37}-11}{3}
The equation is now solved.
3x^{2}+22x+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}+22x+28-28=-28
Subtract 28 from both sides of the equation.
3x^{2}+22x=-28
Subtracting 28 from itself leaves 0.
\frac{3x^{2}+22x}{3}=-\frac{28}{3}
Divide both sides by 3.
x^{2}+\frac{22}{3}x=-\frac{28}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+\frac{22}{3}x+\left(\frac{11}{3}\right)^{2}=-\frac{28}{3}+\left(\frac{11}{3}\right)^{2}
Divide \frac{22}{3}, the coefficient of the x term, by 2 to get \frac{11}{3}. Then add the square of \frac{11}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{22}{3}x+\frac{121}{9}=-\frac{28}{3}+\frac{121}{9}
Square \frac{11}{3} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{22}{3}x+\frac{121}{9}=\frac{37}{9}
Add -\frac{28}{3} to \frac{121}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{11}{3}\right)^{2}=\frac{37}{9}
Factor x^{2}+\frac{22}{3}x+\frac{121}{9}. 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{11}{3}\right)^{2}}=\sqrt{\frac{37}{9}}
Take the square root of both sides of the equation.
x+\frac{11}{3}=\frac{\sqrt{37}}{3} x+\frac{11}{3}=-\frac{\sqrt{37}}{3}
Simplify.
x=\frac{\sqrt{37}-11}{3} x=\frac{-\sqrt{37}-11}{3}
Subtract \frac{11}{3} from both sides of the equation.