Question

How do you solve ${{x}^{2}}+5x+4=0$?

Answer 1

Hint: The equation given in the question is a quadratic equation. A quadratic equation can be solved by applying the quadratic formula, which is given as $x=\dfrac{-b\pm \sqrt{D}}{2a}$, where $D$ is the discriminant. The discriminant $D$ is given by the formula $D={{b}^{2}}-4ac$. In these formulae, $a$ is the coefficient of ${{x}^{2}}$, $b$ is the coefficient of $x$, and $c$ is the constant term in the given quadratic equation. From the quadratic equation given in the question, the respective values of the coefficients are $1$, $5$, and $4$. As is clear from the quadratic formula, we will obtain two values of $x$ which together will contribute to the final solution of the given quadratic equation.

Complete step by step answer:
The given equation is
${{x}^{2}}+5x+4=0.......(i)$
Since the highest power of the variable $x$ in the above equation is $2$, so this is the quadratic equation in $x$.
We know that the solution of the quadratic equation is given by the quadratic formula which is given as
$x=\dfrac{-b\pm \sqrt{D}}{2a}.......(ii)$
So we need to calculate the discriminant of the given quadratic equation. We know that the discriminant is given by
$D={{b}^{2}}-4ac.......(iii)$
From the above equation (i) we note the following values of the coefficients
$\begin{align}
  & \Rightarrow a=1........(iv) \\
 & \Rightarrow b=5........(v) \\
 & \Rightarrow c=4........(vi) \\
\end{align}$
Substituting (iv), (v) and (vi) in (iii), we get
$\Rightarrow D={{\left( 5 \right)}^{2}}-4\left( 1 \right)\left( 4 \right)$
$\Rightarrow D=25-16$
On solving we get
$\Rightarrow D=9.......(vii)$
Now, we substitute (iv), (v) and (vii) in the quadratic formula given in the equation (ii) to get
$\Rightarrow x=\dfrac{-5\pm \sqrt{9}}{2\left( 1 \right)}$
We know that $\sqrt{9}=3$. Substituting this above, we get
$\begin{align}
  & \Rightarrow x=\dfrac{-5\pm 3}{2\left( 1 \right)} \\
 & \Rightarrow x=\dfrac{-5+3}{2\left( 1 \right)}\text{, }x=\dfrac{-5-3}{2\left( 1 \right)} \\
\end{align}$
On solving, we finally get
\[\begin{align}
  & \Rightarrow x=\dfrac{-2}{2},\text{ }x=\dfrac{-8}{2} \\
 & \Rightarrow x=-1,\text{ }x=-4 \\
\end{align}\]
Thus, the values of the variable $x$ which we obtain by solving the above equations are $-1$ and $-4$.
Hence, the solution of the given equation is $x=-1$ and $x=-4$.

Note:
Before applying the quadratic formula in the given quadratic equation, always ensure that it is given in its standard form, that is, the right hand side of the given equation must be zero. If the right hand side is non-zero, then first it has to be made zero and then the quadratic formula has to be applied. Also, this question could also be easily solved by using the middle term split method, by which we could factorise the given equation and obtain the solutions directly.