Question

How do you solve \[{{x}^{2}}+2x=0\]?

Answer 1

Hint: The degree of an equation is the highest power to which the variable is raised. We can decide if the equation is linear, quadratic, cubic, etc. using the degree of the equation. For a quadratic equation \[a{{x}^{2}}+bx+c=0\], using the formula method we can find the roots of the equation as \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\].

Complete step by step answer:
The given equation is \[{{x}^{2}}+2x=0\]. The highest power of the variable in the equation is 2, so the degree of the equation is also 2. It means that the equation is quadratic. Comparing the general equation of the quadratic \[a{{x}^{2}}+bx+c=0\], we get \[a=1,b=2\And c=0\].
We know that using the formula method, we can find the roots of the quadratic as \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]. Substituting the value of the coefficient in this formula, we get
\[\begin{align}
  & \Rightarrow x=\dfrac{-(2)\pm \sqrt{{{\left( 2 \right)}^{2}}-4\left( 1 \right)\left( 0 \right)}}{2(1)} \\
 & \Rightarrow x=\dfrac{-2\pm \sqrt{4}}{2} \\
 & \Rightarrow x=\dfrac{-2\pm 2}{2} \\
\end{align}\]
\[\Rightarrow x=\dfrac{-2+2}{2}\] or \[x=\dfrac{-2-2}{2}\]
\[\Rightarrow x=\dfrac{0}{2}=0\] or \[x=\dfrac{-4}{2}=-2\]
Hence, the roots of the equation are \[x=0\] or \[x=-2\].

Note:
There are many methods to solve a quadratic equation, such as the factorization method, completing square method, and formula method. We can use any of them to solve a quadratic equation. The formula method should be preferred because it gives two roots of the equation, whether they are real or imaginary.
For this question, we have one root without any method. As the constant term in the equation is zero. We can surely say that the equation has one of its roots as 0. Using this information, we can factorize the given equation as follows,
\[\Rightarrow {{x}^{2}}+2x=x\left( x+2 \right)=0\]
Thus, using this factored form we can say the roots of the equation are \[x=0\] or \[x=-2\].