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2x^{2}+3x-2=15
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.
2x^{2}+3x-2-15=15-15
Subtract 15 from both sides of the equation.
2x^{2}+3x-2-15=0
Subtracting 15 from itself leaves 0.
2x^{2}+3x-17=0
Subtract 15 from -2.
x=\frac{-3±\sqrt{3^{2}-4\times 2\left(-17\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 3 for b, and -17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-3±\sqrt{9-4\times 2\left(-17\right)}}{2\times 2}
Square 3.
x=\frac{-3±\sqrt{9-8\left(-17\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-3±\sqrt{9+136}}{2\times 2}
Multiply -8 times -17.
x=\frac{-3±\sqrt{145}}{2\times 2}
Add 9 to 136.
x=\frac{-3±\sqrt{145}}{4}
Multiply 2 times 2.
x=\frac{\sqrt{145}-3}{4}
Now solve the equation x=\frac{-3±\sqrt{145}}{4} when ± is plus. Add -3 to \sqrt{145}.
x=\frac{-\sqrt{145}-3}{4}
Now solve the equation x=\frac{-3±\sqrt{145}}{4} when ± is minus. Subtract \sqrt{145} from -3.
x=\frac{\sqrt{145}-3}{4} x=\frac{-\sqrt{145}-3}{4}
The equation is now solved.
2x^{2}+3x-2=15
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}+3x-2-\left(-2\right)=15-\left(-2\right)
Add 2 to both sides of the equation.
2x^{2}+3x=15-\left(-2\right)
Subtracting -2 from itself leaves 0.
2x^{2}+3x=17
Subtract -2 from 15.
\frac{2x^{2}+3x}{2}=\frac{17}{2}
Divide both sides by 2.
x^{2}+\frac{3}{2}x=\frac{17}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=\frac{17}{2}+\left(\frac{3}{4}\right)^{2}
Divide \frac{3}{2}, the coefficient of the x term, by 2 to get \frac{3}{4}. Then add the square of \frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{17}{2}+\frac{9}{16}
Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{145}{16}
Add \frac{17}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{3}{4}\right)^{2}=\frac{145}{16}
Factor x^{2}+\frac{3}{2}x+\frac{9}{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{3}{4}\right)^{2}}=\sqrt{\frac{145}{16}}
Take the square root of both sides of the equation.
x+\frac{3}{4}=\frac{\sqrt{145}}{4} x+\frac{3}{4}=-\frac{\sqrt{145}}{4}
Simplify.
x=\frac{\sqrt{145}-3}{4} x=\frac{-\sqrt{145}-3}{4}
Subtract \frac{3}{4} from both sides of the equation.