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2x^{2}+7x=84
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}+7x-84=84-84
Subtract 84 from both sides of the equation.
2x^{2}+7x-84=0
Subtracting 84 from itself leaves 0.
x=\frac{-7±\sqrt{7^{2}-4\times 2\left(-84\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 7 for b, and -84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-7±\sqrt{49-4\times 2\left(-84\right)}}{2\times 2}
Square 7.
x=\frac{-7±\sqrt{49-8\left(-84\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-7±\sqrt{49+672}}{2\times 2}
Multiply -8 times -84.
x=\frac{-7±\sqrt{721}}{2\times 2}
Add 49 to 672.
x=\frac{-7±\sqrt{721}}{4}
Multiply 2 times 2.
x=\frac{\sqrt{721}-7}{4}
Now solve the equation x=\frac{-7±\sqrt{721}}{4} when ± is plus. Add -7 to \sqrt{721}.
x=\frac{-\sqrt{721}-7}{4}
Now solve the equation x=\frac{-7±\sqrt{721}}{4} when ± is minus. Subtract \sqrt{721} from -7.
x=\frac{\sqrt{721}-7}{4} x=\frac{-\sqrt{721}-7}{4}
The equation is now solved.
2x^{2}+7x=84
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{2x^{2}+7x}{2}=\frac{84}{2}
Divide both sides by 2.
x^{2}+\frac{7}{2}x=\frac{84}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+\frac{7}{2}x=42
Divide 84 by 2.
x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=42+\left(\frac{7}{4}\right)^{2}
Divide \frac{7}{2}, the coefficient of the x term, by 2 to get \frac{7}{4}. Then add the square of \frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{7}{2}x+\frac{49}{16}=42+\frac{49}{16}
Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{721}{16}
Add 42 to \frac{49}{16}.
\left(x+\frac{7}{4}\right)^{2}=\frac{721}{16}
Factor x^{2}+\frac{7}{2}x+\frac{49}{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{7}{4}\right)^{2}}=\sqrt{\frac{721}{16}}
Take the square root of both sides of the equation.
x+\frac{7}{4}=\frac{\sqrt{721}}{4} x+\frac{7}{4}=-\frac{\sqrt{721}}{4}
Simplify.
x=\frac{\sqrt{721}-7}{4} x=\frac{-\sqrt{721}-7}{4}
Subtract \frac{7}{4} from both sides of the equation.