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How do you solve $3{x^2} + 6x = 8$?

seo-qna
Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint: We are given a quadratic algebraic expression of the form $a{x^2} + bx + c$ it is a quadratic expression. And we have to find its factor. Here the first term is $a{x^2}$ its coefficient is$a$. The middle term is $ - bx$. Coefficient is $b$. The last term is constant i.e. $ - c$. First we will equate the equation equal to zero then by using the quadratic formula we find the value of $x$. Quadratic formula states:

Formula used:
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Here $b$= coefficient of $x$, coefficient of\[\;{x^2}\]and\[\;c\]= constant term. After substituting the values in the formula we will get two values of\[x\].

Complete step by step answer:
Given a quadratic equation and we have to find the value of \[x\]. The given quadratic equation is $3{x^2} + 6x = 8$ so first we will equate the equation equal to zero. For this subtract \[8\] from both sides we will get:
$ \Rightarrow 3{x^2} + 6x - 8 = 8 - 8$
$ \Rightarrow 3{x^2} + 6x - 8 = 0$

Using the quadratic formula we will get the value of x.
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Here $a = 3,b = - 6,c = - 8$. Substituting these values in the equation we will get the value of \[x\].
$ \Rightarrow \dfrac{{6 \pm \sqrt {{{\left( { - 6} \right)}^2} - 4.(3 \times 8)} }}{{2 \times 3}}$
Now we will raise $ - 6$ to the power of $2$.
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {36 - 4.\left( {3. - 8} \right)} }}{{2 \times 3}}$
Now we will multiply $3$ by \[ - 8\] and open the bracket by multiplying it by\[ - 4\].
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {36 - 4.\left( { - 24} \right)} }}{{2 \times 3}}$
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {36 + 96} }}{{2 \times 3}}$

Now on adding \[36\] and \[\;96\].
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {132} }}{{2 \times 3}}$
Now we will rewrite \[132\]as${2^2} \times 33$
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {{2^2} \times 33} }}{{2 \times 3}}$
Now we will pull out the terms ${2^2}$ under the radical. Also solve the denominator.
$ \Rightarrow x = \dfrac{{6 \pm 2\sqrt {33} }}{6}$
On simplifying the expression we will get:
$ \Rightarrow x = \dfrac{{3 \pm \sqrt {33} }}{3}$
Converting these in decimal form we will get:
$x = 4.917; x = - 1.083$
Hence the value of x is $x = 4.917;x = - 1.083$.

Note: In such types of questions if we are given a quadratic equation then it can be easily solved by splitting the middle term. If the equation is a simple form but if the equation contains big coefficients or any such coefficients that cannot be solved by splitting the middle term. Then use the quadratic formula i.e.$\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ by using this we can solve any complicated equation. But for small simple equation then we will use the splitting the middle term method for this also there are two types of equation form for $a{x^2} + bx + c$ we will split the middle term in form of addition and for $a{x^2} + bx - c$. We will split the middle term in form subtraction. Remember this while solving by splitting the middle term.