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4x^{2}+12x=5
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.
4x^{2}+12x-5=5-5
Subtract 5 from both sides of the equation.
4x^{2}+12x-5=0
Subtracting 5 from itself leaves 0.
x=\frac{-12±\sqrt{12^{2}-4\times 4\left(-5\right)}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 12 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-12±\sqrt{144-4\times 4\left(-5\right)}}{2\times 4}
Square 12.
x=\frac{-12±\sqrt{144-16\left(-5\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-12±\sqrt{144+80}}{2\times 4}
Multiply -16 times -5.
x=\frac{-12±\sqrt{224}}{2\times 4}
Add 144 to 80.
x=\frac{-12±4\sqrt{14}}{2\times 4}
Take the square root of 224.
x=\frac{-12±4\sqrt{14}}{8}
Multiply 2 times 4.
x=\frac{4\sqrt{14}-12}{8}
Now solve the equation x=\frac{-12±4\sqrt{14}}{8} when ± is plus. Add -12 to 4\sqrt{14}.
x=\frac{\sqrt{14}-3}{2}
Divide -12+4\sqrt{14} by 8.
x=\frac{-4\sqrt{14}-12}{8}
Now solve the equation x=\frac{-12±4\sqrt{14}}{8} when ± is minus. Subtract 4\sqrt{14} from -12.
x=\frac{-\sqrt{14}-3}{2}
Divide -12-4\sqrt{14} by 8.
x=\frac{\sqrt{14}-3}{2} x=\frac{-\sqrt{14}-3}{2}
The equation is now solved.
4x^{2}+12x=5
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.
\frac{4x^{2}+12x}{4}=\frac{5}{4}
Divide both sides by 4.
x^{2}+\frac{12}{4}x=\frac{5}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}+3x=\frac{5}{4}
Divide 12 by 4.
x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\frac{5}{4}+\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}=\frac{5+9}{4}
Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+3x+\frac{9}{4}=\frac{7}{2}
Add \frac{5}{4} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{3}{2}\right)^{2}=\frac{7}{2}
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{7}{2}}
Take the square root of both sides of the equation.
x+\frac{3}{2}=\frac{\sqrt{14}}{2} x+\frac{3}{2}=-\frac{\sqrt{14}}{2}
Simplify.
x=\frac{\sqrt{14}-3}{2} x=\frac{-\sqrt{14}-3}{2}
Subtract \frac{3}{2} from both sides of the equation.