Question

How do you solve $3{x^2} + 11x - 4 = 0$ by factoring?

Answer 1

Hint: In this question we need to solve for the given quadratic equation $3{x^2} + 11x - 4 = 0$ by factor method of finding roots of quadratic equation. In factor method finding roots we split term of $x$ to get the factor of given quadratic equation and its roots. We can also find the roots of quadratic equations also by quadratic formula.

Complete step by step answer:
Let us try to find the root of a given quadratic equation $3{x^2} + 11x - 4 = 0$ by factor method of finding roots of quadratic equation. In factor method we for any quadratic equation $a{x^2} + bx + c = 0$ term with $x$ such that $bx = {b_1}x + {b_2}x$ and the product of the coefficients of $x$ are equal to the product of $a$ coefficient of ${x^2}$ and the constant $c$. After which we took common terms and made them equal to $0$, finally to get the root of the quadratic equation.
Let us now apply the procedure of factor method to find the root of a given quadratic equation$3{x^2} + 11x - 4 = 0$.
For given quadratic equation we have,
$
\Rightarrow a = 3 \\
\Rightarrow b = 11 \\
\Rightarrow c = - 4 \\
 $
Now, the product of $a$ and $b$
$ \Rightarrow a \times b = 3 \times ( - 4) = - 12$

Now, we have to split $b$ such that $b = {b_1} + {b_2}$ and ${b_1} \times {b_2} = - 12$.
Two numbers satisfying both the conditions are $12$ and $ - 1$. Since $12 + ( - 1) = 11 = b$ and $12 \times ( - 1) = - 12 = a \times b$.
 So, given quadratic equation can be written as
$ \Rightarrow 3{x^2} + 11x - 4 = 0$
$ \Rightarrow 3{x^2} + 12x - x - 4 = 0$
 Now combining terms, we get
$ \Rightarrow (3{x^2} + 12x) + ( - x - 4) = 0$

Now taking out $3x$ from $(3{x^2} + 12x)$ and $ - 1$ from $( - x - 4)$we get,
$ \Rightarrow 3x(x + 4) - 1(x + 4) = 0$
Now taking $(x + 4)$common from above equation we get,
$ \Rightarrow (x + 4)(3x - 1) = 0$
Now assuming both terms are equal to $0$ from the above equation we get the root of the given quadratic equation.
$
\Rightarrow x + 4 = 0 \\
\Rightarrow x = - 4 \\
 $
And, $
\Rightarrow 3x - 1 = 0 \\
\Rightarrow 3x = 1 \\
\Rightarrow x = \dfrac{{ - 1}}{3} \\ $
Hence the root of the quadratic equation $3{x^2} + 11x - 4 = 0$ are $x = - 4$ and $x = \dfrac{{ - 1}}{3}$.

Note: In questions where we are asked to solve for the root of the quadratic equation by factor method we need to be careful while we are splitting coefficients of $x$ and about the signs of terms. We can also find the root of the quadratic formula and complete the square method. Quadratic formula is the easiest method to find the root of a quadratic equation.