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x^{2}+4x=\frac{27}{4}
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}+4x-\frac{27}{4}=\frac{27}{4}-\frac{27}{4}
Subtract \frac{27}{4}=6.75 from both sides of the equation.
x^{2}+4x-\frac{27}{4}=0
Subtracting \frac{27}{4}=6.75 from itself leaves 0.
x=\frac{-4±\sqrt{4^{2}-4\left(-\frac{27}{4}\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -\frac{27}{4}=-6.75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\left(-\frac{27}{4}\right)}}{2}
Square 4.
x=\frac{-4±\sqrt{16+27}}{2}
Multiply -4 times -\frac{27}{4}=-6.75.
x=\frac{-4±\sqrt{43}}{2}
x=\frac{\sqrt{43}-4}{2}
Now solve the equation x=\frac{-4±\sqrt{43}}{2} when ± is plus. Add -4 to \sqrt{43}\approx 6.557438524.
x=\frac{\sqrt{43}}{2}-2
Divide -4+\sqrt{43}\approx 2.557438524 by 2.
x=\frac{-\sqrt{43}-4}{2}
Now solve the equation x=\frac{-4±\sqrt{43}}{2} when ± is minus. Subtract \sqrt{43}\approx 6.557438524 from -4.
x=-\frac{\sqrt{43}}{2}-2
Divide -4-\sqrt{43}\approx -10.557438524 by 2.
x=\frac{\sqrt{43}}{2}-2 x=-\frac{\sqrt{43}}{2}-2
The equation is now solved.
x^{2}+4x=\frac{27}{4}
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}+4x+2^{2}=\frac{27}{4}+2^{2}
Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+4x+4=\frac{27}{4}+4
Square 2.
x^{2}+4x+4=\frac{43}{4}