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x\left(2x+7\right)=0
Factor out x.
x=0 x=-\frac{7}{2}
To find equation solutions, solve x=0 and 2x+7=0.
2x^{2}+7x=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{-7±\sqrt{7^{2}}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 7 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-7±7}{2\times 2}
Take the square root of 7^{2}.
x=\frac{-7±7}{4}
Multiply 2 times 2.
x=\frac{0}{4}
Now solve the equation x=\frac{-7±7}{4} when ± is plus. Add -7 to 7.
x=0
Divide 0 by 4.
x=-\frac{14}{4}
Now solve the equation x=\frac{-7±7}{4} when ± is minus. Subtract 7 from -7.
x=-\frac{7}{2}
Reduce the fraction \frac{-14}{4} to lowest terms by extracting and canceling out 2.
x=0 x=-\frac{7}{2}
The equation is now solved.
2x^{2}+7x=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.
\frac{2x^{2}+7x}{2}=\frac{0}{2}
Divide both sides by 2.
x^{2}+\frac{7}{2}x=\frac{0}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+\frac{7}{2}x=0
Divide 0 by 2.
x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=\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}=\frac{49}{16}
Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.
\left(x+\frac{7}{4}\right)^{2}=\frac{49}{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{49}{16}}
Take the square root of both sides of the equation.
x+\frac{7}{4}=\frac{7}{4} x+\frac{7}{4}=-\frac{7}{4}
Simplify.
x=0 x=-\frac{7}{2}
Subtract \frac{7}{4} from both sides of the equation.