Question

How do you solve \[{p^2} + 3p - 9 = 0\] by completing the square method ?

Answer 1

Hint: The given problem requires us to solve an equation given to us in the question itself \[{p^2} + 3p - 9 = 0\] using completing the square method. There are various methods that can be employed to solve a quadratic equation like completing the square method, using quadratic formulas and by splitting the middle term. Completing the square method involves manipulating the equation to form a perfect square and solving for variable or unknown.

Complete step-by-step answer:
In the given question, we have to solve the given equation \[{p^2} + 3p - 9 = 0\] using the completing the square method.
So, \[{p^2} + 3p - 9 = 0\]
So, for completing the square, we have to make the given equation resemble the whole square of a binomial identity: ${\left( {a + b} \right)^2} = \left( {{a^2} + 2ab + {b^2}} \right)$. So, we get,
 \[ \Rightarrow {\left( p \right)^2} + 2\left( {\dfrac{3}{2}} \right)\left( p \right) - 9 = 0\]
Adding square of the term \[\left( {\dfrac{3}{2}} \right)\] so as to exactly match the above mentioned identity,
 \[ \Rightarrow {\left( p \right)^2} + 2\left( {\dfrac{3}{2}} \right)\left( p \right) + {\left( {\dfrac{3}{2}} \right)^2} - {\left( {\dfrac{3}{2}} \right)^2} - 9 = 0\]
So, the first three terms of the equation resemble the identity ${\left( {a + b} \right)^2} = \left( {{a^2} + 2ab + {b^2}} \right)$. Hence, \[\left[ {{{\left( p \right)}^2} + 2\left( {\dfrac{3}{2}} \right)\left( p \right) + {{\left( {\dfrac{3}{2}} \right)}^2}} \right] \] can be condensed as \[{\left( {p + \dfrac{3}{2}} \right)^2}\] . So, we get,
 \[ \Rightarrow {\left( {p + \dfrac{3}{2}} \right)^2} - {\left( {\dfrac{3}{2}} \right)^2} - 9 = 0\]
Shifting the constant terms to right side of the equation, we get,
 \[ \Rightarrow {\left( {p + \dfrac{3}{2}} \right)^2} = \dfrac{9}{4} + 9\]
Simplifying the right side of the equation, we get,
 \[ \Rightarrow {\left( {p + \dfrac{3}{2}} \right)^2} = \dfrac{{45}}{4}\]
Taking square root in both sides of the equation, we get,
 \[ \Rightarrow \left( {p + \dfrac{3}{2}} \right) = \pm \sqrt {\dfrac{{45}}{4}} \]
Simplifying the right side of the equation, we get,
 \[ \Rightarrow \left( {p + \dfrac{3}{2}} \right) = \pm \dfrac{{3\sqrt 5 }}{2}\]
Shifting the term $\dfrac{3}{2}$ to right side of the equation, we get,
 \[ \Rightarrow p = \dfrac{{ - 3 \pm 3\sqrt 5 }}{2}\]
So, the roots of the equation \[{p^2} + 3p - 9 = 0\] are : \[p = \dfrac{{ - 3 + 3\sqrt 5 }}{2}\] and \[p = \dfrac{{ - 3 - 3\sqrt 5 }}{2}\] .
So, the correct answer is “ \[p = \dfrac{{ - 3 + 3\sqrt 5 }}{2}\] and \[p = \dfrac{{ - 3 - 3\sqrt 5 }}{2}\] ”.

Note: Quadratic equations are the polynomial equations with degree of the variable or unknown as $2$. Quadratic equations can be solved by splitting the middle term, factoring common factors, using the quadratic formula and completing the square method. The given equation can be solved by each and every method listed above.