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x^{2}+3x+4=27
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^{2}+3x+4-27=27-27
Subtract 27 from both sides of the equation.
x^{2}+3x+4-27=0
Subtracting 27 from itself leaves 0.
x^{2}+3x-23=0
Subtract 27 from 4.
x=\frac{-3±\sqrt{3^{2}-4\left(-23\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 3 for b, and -23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-3±\sqrt{9-4\left(-23\right)}}{2}
Square 3.
x=\frac{-3±\sqrt{9+92}}{2}
Multiply -4 times -23.
x=\frac{-3±\sqrt{101}}{2}
Add 9 to 92.
x=\frac{\sqrt{101}-3}{2}
Now solve the equation x=\frac{-3±\sqrt{101}}{2} when ± is plus. Add -3 to \sqrt{101}.
x=\frac{-\sqrt{101}-3}{2}
Now solve the equation x=\frac{-3±\sqrt{101}}{2} when ± is minus. Subtract \sqrt{101} from -3.
x=\frac{\sqrt{101}-3}{2} x=\frac{-\sqrt{101}-3}{2}
The equation is now solved.
x^{2}+3x+4=27
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+4-4=27-4
Subtract 4 from both sides of the equation.
x^{2}+3x=27-4
Subtracting 4 from itself leaves 0.
x^{2}+3x=23
Subtract 4 from 27.
x^{2}+3x+\left(\frac{3}{2}\right)^{2}=23+\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}=23+\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{101}{4}
Add 23 to \frac{9}{4}.
\left(x+\frac{3}{2}\right)^{2}=\frac{101}{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{101}{4}}
Take the square root of both sides of the equation.
x+\frac{3}{2}=\frac{\sqrt{101}}{2} x+\frac{3}{2}=-\frac{\sqrt{101}}{2}
Simplify.
x=\frac{\sqrt{101}-3}{2} x=\frac{-\sqrt{101}-3}{2}
Subtract \frac{3}{2} from both sides of the equation.