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2x^{2}+9x-2.5=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{-9±\sqrt{9^{2}-4\times 2\left(-2.5\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 9 for b, and -2.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-9±\sqrt{81-4\times 2\left(-2.5\right)}}{2\times 2}
Square 9.
x=\frac{-9±\sqrt{81-8\left(-2.5\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-9±\sqrt{81+20}}{2\times 2}
Multiply -8 times -2.5.
x=\frac{-9±\sqrt{101}}{2\times 2}
Add 81 to 20.
x=\frac{-9±\sqrt{101}}{4}
Multiply 2 times 2.
x=\frac{\sqrt{101}-9}{4}
Now solve the equation x=\frac{-9±\sqrt{101}}{4} when ± is plus. Add -9 to \sqrt{101}.
x=\frac{-\sqrt{101}-9}{4}
Now solve the equation x=\frac{-9±\sqrt{101}}{4} when ± is minus. Subtract \sqrt{101} from -9.
x=\frac{\sqrt{101}-9}{4} x=\frac{-\sqrt{101}-9}{4}
The equation is now solved.
2x^{2}+9x-2.5=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.
2x^{2}+9x-2.5-\left(-2.5\right)=-\left(-2.5\right)
Add 2.5 to both sides of the equation.
2x^{2}+9x=-\left(-2.5\right)
Subtracting -2.5 from itself leaves 0.
2x^{2}+9x=2.5
Subtract -2.5 from 0.
\frac{2x^{2}+9x}{2}=\frac{2.5}{2}
Divide both sides by 2.
x^{2}+\frac{9}{2}x=\frac{2.5}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+\frac{9}{2}x=1.25
Divide 2.5 by 2.
x^{2}+\frac{9}{2}x+\left(\frac{9}{4}\right)^{2}=1.25+\left(\frac{9}{4}\right)^{2}
Divide \frac{9}{2}, the coefficient of the x term, by 2 to get \frac{9}{4}. Then add the square of \frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{9}{2}x+\frac{81}{16}=1.25+\frac{81}{16}
Square \frac{9}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{9}{2}x+\frac{81}{16}=\frac{101}{16}
Add 1.25 to \frac{81}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{9}{4}\right)^{2}=\frac{101}{16}
Factor x^{2}+\frac{9}{2}x+\frac{81}{16}. 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{9}{4}\right)^{2}}=\sqrt{\frac{101}{16}}
Take the square root of both sides of the equation.
x+\frac{9}{4}=\frac{\sqrt{101}}{4} x+\frac{9}{4}=-\frac{\sqrt{101}}{4}
Simplify.
x=\frac{\sqrt{101}-9}{4} x=\frac{-\sqrt{101}-9}{4}
Subtract \frac{9}{4} from both sides of the equation.