Question

How do you factor $3{x^2} + 10x - 8 = 0$

Answer 1

Hint: First compare the given quadratic equation with the standard equation $a{x^2} + bx + c = 0$ then find the two integers whose sum is equals to $b$ and whose product is equal to $\left( {ac} \right)$. Then calculate the factors of the given equation.

Complete step by step solution:
The given equation is
$3{x^2} + 10x - 8 = 0 \cdots \cdots \left( 1 \right)$
The standard quadratic equation is
$a{x^2} + bx + c = 0 \cdots \cdots \left( 2 \right)$
Now compare equations (1) and (2) to get the values of variables $a,b$ and $c$.
Therefore, the required variables are
$\
  a = 3 \\
  b = 10 \\
  c = - 8 \\
\ $
Now we have to find out the pair of integers whose product is equal to $ - 24$ and sum is equal to $10$.
In that case the required pair is $\left( {12, - 2} \right)$.
Therefore, the equation (1) can be written in the following form:
$3{x^2} + 12x - 2x - 8 = 0$
Now further solving
$3x\left( {x + 4} \right) - 2\left( {x + 4} \right) = 0$
$\boxed{ \Rightarrow \left( {3x - 2} \right)\left( {x + 4} \right) = 0}$
Hence, the factors of the given equation $3{x^2} + 10x - 8 = 0$ are $\left( {3x - 2} \right)\left( {x + 4} \right) = 0$.
Here we are seeing that the product of two terms is equal to zero.
When the product of several terms becomes zero it means at least one term of the two terms is zero. Now we have to solve these two terms by equating zero separately.
Now first
$\left( {3x - 2} \right) = 0$
$ \Rightarrow x = \dfrac{2}{3}$
And similarly,
$\left( {x + 4} \right) = 0$
$ \Rightarrow x = - 4$

Therefore, the solutions of the given quadratic equation are
$x = - \dfrac{2}{3},4$

Note:
Solving the quadratic equation by parts is the simplest method in mathematics. In this method we only have to pay attention while choosing factors of the given equation. While doing calculations please take the case of the sign of the numbers. These types of equations can be solved by the Sridharacharya method and the completing square method.