Question

How do you solve $3x(x+2)=2$ using the quadratic formula?

Answer 1

Hint: In this question, we have to find the value of x. Thus, we will use the discriminant quadratic formula to get the solution. First, we will apply the distributive property $a(b+c)=ab+ac$ on the left-hand side of the equation, and then subtract 2 on both sides of the equation. After the necessary calculations, we compare the general form of quadratic equation and the new quadratic equation to get the value of a, b, and c. Then, we will get the value of discriminant $D=\sqrt{{{b}^{2}}-4ac}$, and thus find the value of x using the discriminant formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .After necessary calculations, get two equations of x, we solve them separately to get the value of x, which is our required answer.

Complete step by step answer:
According to the question, we have to find the value of x from a quadratic equation,
Thus, we will apply the discriminant formula to get the solution.
The equation to be solved is $3x(x+2)=2$ --------- (1)
Now, we will apply the distributive property $a(b-c)=ab-ac$ on the left-hand side in the equation (1), we get
$\Rightarrow 3{{x}^{2}}+3x\times 2=2$
On further solving, we get
$\Rightarrow 3{{x}^{2}}+6x=2$
Now, we will subtract 2 on both sides in the above equation, we get
\[\Rightarrow 3{{x}^{2}}+6x-2=2-2\]
As we know, the same terms with opposite signs cancel out each other, therefore we get
$\Rightarrow 3{{x}^{2}}+6x-2=0$ ----------- (2)
As we know, the general quadratic equation is in form of $a{{x}^{2}}+bx+c=0$ ---------- (3)
Thus, on comparing equation (2) and (3), we get $a=3,$ $b=6,$ and $c=-2$ ------- (4)
So, now we will apply the discriminant formula $D=\sqrt{{{b}^{2}}-4ac}$ by putting the values of equation (4) in the formula, we get
$\begin{align}
  & \Rightarrow D=\sqrt{{{(6)}^{2}}-4.(3).(-2)} \\
 & \Rightarrow D=\sqrt{36+24} \\
\end{align}$
Thus, on further solving, we get
$\Rightarrow D=\sqrt{60}$ ---------- (5)
Since D>0 implies it has the real roots.
Since we see the discriminate is areal number, thus now we will find the value of x, using the formula,
$\Rightarrow x=\dfrac{-b\pm D}{2a}$
$\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ --------------- (6)
So, we will put the value of equation (4) and (5) in equation (6), we get
$\Rightarrow x=\dfrac{-(6)\pm \sqrt{60}}{2.(3)}$
On further simplification, we get
$\Rightarrow x=\dfrac{-6\pm 2\sqrt{15}}{6}$
Therefore, we will split the above equation in terms of (+) and (-), we get
$\Rightarrow x=\dfrac{-6+2\sqrt{15}}{6}$ -------- (7) , or
 $\Rightarrow x=\dfrac{-6-2\sqrt{15}}{6}$ ---------- (8)
Now, we will first solve equation (7), we get
$\Rightarrow x=\dfrac{-6}{6}+\dfrac{2\sqrt{15}}{6}$
On further simplification, we get
$\Rightarrow x=-1+\dfrac{\sqrt{15}}{3}$
Now, we will first solve equation (8), we get
$\Rightarrow x=\dfrac{-6}{6}-\dfrac{2\sqrt{15}}{6}$
On further simplification, we get
$\Rightarrow x=-1-\dfrac{\sqrt{15}}{3}$

Therefore, for the equation $3x(x+2)=2$ , we get the value of\[x=-1+\dfrac{\sqrt{15}}{3},-1-\dfrac{\sqrt{15}}{3}\]

Note: While solving this problem, do all the steps carefully and avoid mathematical errors to get the correct answer. Do mention all the formulas and the property you are using to get an accurate answer.