Question

Solve the following equation using the formula method:
\[7{{p}^{2}}-5p=2\]

Answer 1

Hint: In this question, we first need to rearrange the terms in the equation given in the question. Then we can apply the direct formula to the quadratic equation and further simplify it to get the values of p that satisfies the given quadratic equation.

Complete step-by-step answer:
Now, from the given equation in the question we have
\[7{{p}^{2}}-5p=2\]
Quadratic Polynomial:
A polynomial of second degree is called a quadratic polynomial.
A quadratic polynomial when equated to zero is called a quadratic equation.
The general form of a quadratic equation is given by
\[a{{x}^{2}}+bx+c=0,a\ne 0\]
As we already know that a quadratic equation has two solutions which can be found by using the direct formula.
Now, the direct formula is given by
\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Now, from the given quadratic equation we have
\[\Rightarrow 7{{p}^{2}}-5p=2\]
Let us now rearrange the terms in the above equation
\[\Rightarrow 7{{p}^{2}}-5p-2=0\]
Now, on comparing this with the general form of a quadratic equation we get,
\[\begin{align}
  & a=7 \\
 & b=-5 \\
 & c=-2 \\
\end{align}\]
Now, let us substitute these values in the direct formula
\[\Rightarrow p=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Now, on substituting the respective values we get,
\[\Rightarrow p=\dfrac{-\left( -5 \right)\pm \sqrt{{{\left( -5 \right)}^{2}}-4\times 7\times \left( -2 \right)}}{2\times 7}\]
Now, this can also be written as
\[\Rightarrow p=\dfrac{5\pm \sqrt{25+56}}{14}\]
Now, on further simplification we get,
\[\Rightarrow p=\dfrac{5\pm 9}{14}\]
Now, on considering both the conditions we get,
\[\Rightarrow p=\dfrac{5+9}{14}\]
Now, on simplifying it further we get,
\[\begin{align}
  & \Rightarrow p=\dfrac{14}{14} \\
 & \therefore p=1 \\
\end{align}\]
Let us now consider the other possible case
\[\Rightarrow p=\dfrac{5-9}{14}\]
Now, on further simplification we get,
\[\begin{align}
  & \Rightarrow p=\dfrac{-4}{14} \\
 & \therefore p=\dfrac{-2}{7} \\
\end{align}\]
Hence, the values of p which satisfy the given equation are \[1,\dfrac{-2}{7}\]

Note: Instead of using the direct formula we can also find it by using the factorisation method. But as asked in the question to solve using the direct formula we need to use that particular method. However, we can verify the result by using the factorisation method.
It is important to note that while rearranging the terms or when substituting the values in the formula we should not neglect any of the terms or substitute incorrectly because it changes the result.