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Question: How do you solve the quadratic equation \(3{x^2} + 9x = 12\)?...

How do you solve the quadratic equation 3x2+9x=123{x^2} + 9x = 12?

Explanation

Solution

First bring the 12 from the RHS to the LHS and then take out 3 common and cross it off. Then, use the method of “splitting the middle term” to make the factors and thus we have the roots.

Complete step-by-step solution:
We are given that we are required to solve 3x2+9x=123{x^2} + 9x = 12.
Taking 12 from addition in the right hand side to subtraction in the left hand side of the above mentioned equation, we will then obtain the following equation with us:-
⇒3x2+9x−12=0\Rightarrow 3{x^2} + 9x - 12 = 0
Taking 3 common and crossing it off from the left hand side of the above equation, we will then obtain the following equation with us:-
⇒x2+3x−4=0\Rightarrow {x^2} + 3x - 4 = 0
We can write the above mentioned equation as follows:-
⇒x2−x+4x−4=0\Rightarrow {x^2} - x + 4x - 4 = 0
Taking x common from the first two terms from the above mentioned equation, we will then obtain the following equation with us:-
⇒x(x−1)+4x−4=0\Rightarrow x\left( {x - 1} \right) + 4x - 4 = 0
Taking 4 common from the last two terms from the above mentioned equation, we will then obtain the following equation with us:-
⇒x(x−1)+4(x−1)=0\Rightarrow x\left( {x - 1} \right) + 4\left( {x - 1} \right) = 0
Taking (x – 1) common from the last two terms in the latter factor, we will then obtain the following equation with us:-
⇒(x−1)(x+4)=0\Rightarrow \left( {x - 1} \right)\left( {x + 4} \right) = 0
Thus, we have the required roots as 1 and – 4.

Note: The students must notice that we have an alternate way of factoring the quadratic equation involved in it as well. The alternate way is as follows:-
The given equation is x2+3x−4=0{x^2} + 3x - 4 = 0.
Using the quadratic formula given by if the equation is given by ax2+bx+c=0a{x^2} + bx + c = 0, its roots are given by the following equation:-
⇒x=−b±b2−4ac2a\Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}
Thus, we have the roots of x2+3x−4=0{x^2} + 3x - 4 = 0 given by:
⇒x=−3±(3)2−4×(−4)2\Rightarrow x = \dfrac{{ - 3 \pm \sqrt {{{(3)}^2} - 4 \times ( - 4)} }}{2}
Simplifying the calculations in the square root in the numerator of the right hand side, we will then obtain the following equation with us:-
⇒x=−3±9+162\Rightarrow x = \dfrac{{ - 3 \pm \sqrt {9 + 16} }}{2}
Simplifying the calculations in the square root in the numerator of the right hand side further, we will then obtain the following equation with us:-
⇒x=−3±52\Rightarrow x = \dfrac{{ - 3 \pm 5}}{2}
Hence, the roots are 1 and - 4.