Question

How do you find all the zeros of the function $f\left( x \right)=4{{x}^{2}}+29x+30$?

Answer 1

Hint: For this problem we need to calculate the roots or zeros of the given function. We can observe that the given function is a quadratic equation. We have so many methods which are used to calculate the roots of the quadratic equation like completing squares, factorization, and quadratic formula. For this particular problem we are going to use the quadratic formula method. This method says that the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ are given by $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$. So first we will compare the given equation with $a{{x}^{2}}+bx+c=0$ and use the formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ to get the roots of the given equation.

Complete step by step solution:
Given that, $f\left( x \right)=4{{x}^{2}}+29x+30$.
Comparing the above quadratic equation with standard form of the quadratic equation which is $a{{x}^{2}}+bx+c=0$, then we will get
$a=4$, $b=29$, $c=30$.
The method quadratic formula says that the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ are given by $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$. From this formula the roots of the given quadratic equation can be calculated as
$\Rightarrow x=\dfrac{-\left( 29 \right)\pm \sqrt{{{\left( 29 \right)}^{2}}-4\left( 4 \right)\left( 30 \right)}}{2\left( 4 \right)}$
Simplifying the above equation, then we will get
$\begin{align}
  & \Rightarrow x=\dfrac{-29\pm \sqrt{841-480}}{8} \\
 & \Rightarrow x=\dfrac{-29\pm \sqrt{361}}{8} \\
\end{align}$
In the above equation, we have the value $\sqrt{361}$. We can write the value $361$ as $19\times 19$. Now the value $\sqrt{361}$ will be $\sqrt{361}=\sqrt{{{19}^{2}}}=19$. Substituting this value in the above equation, then we will get
$\begin{align}
  & \Rightarrow x=\dfrac{-29\pm 19}{8} \\
 & \Rightarrow x=\dfrac{-29+19}{8}\text{ or }\dfrac{-29-19}{8} \\
 & \Rightarrow x=\dfrac{-10}{8}\text{ or }\dfrac{-48}{8} \\
 & \Rightarrow x=-\dfrac{5}{4}\text{ or }-6 \\
\end{align}$
Hence the roots of the given equation $f\left( x \right)=4{{x}^{2}}+29x+30$ are $x=-6,-\dfrac{5}{4}$.

Note:
We can also calculate the roots of the given equation by plotting the graph of the given equation. When we plot the graph of the given equation in coordinate system then we will get

From the above graph also, we can say that the roots of the equation as $x=-6,-\dfrac{5}{4}$.