Question

How do you factor $-6{{x}^{2}}+31x-35$ ?

Answer 1

Hint: The given equation is a quadratic form. To find the factors for a quadratic equation we can use the formula or we can simplify the equation and use the sum-product pattern method to solve the equation. Here the given equation has a negative sign. So in the first step, we need to solve it and simplify the equation.

Complete step-by-step solution:
The given equation is $-6{{x}^{2}}+31x-35$. For this equation, we need to find out the factors.
As the equation is quadratic we can easily simplify the equation to get the factors.
 In the first step let’s take the common factor from the three terms. Here the common factor is -1.
\[\Rightarrow -6{{x}^{2}}+31x-35\]
\[\Rightarrow -1\left( 6{{x}^{2}}-31x+35 \right)\]
Now we need to simplify the above equation by using the sum-product pattern.
We all know that in the sum-product pattern one term is divided into two-term without changing the value of the equation.
Now we do the same process to the middle term of the equation.
Here the middle term of the equation $-6{{x}^{2}}+31x-35$ is 31x.
The modified equation will be
\[\Rightarrow -1\left( 6{{x}^{2}}-31x+35 \right)\]
$\Rightarrow -1\left( 6{{x}^{2}}-10x-21x+35 \right)$
In the modified equation there are four terms in the bracket.
To find out the factors let’s take out one common factor from the first two terms and another common factor from the next two terms.
$\Rightarrow -1\left( 2x\left( 3x-5 \right)-7\left( 3x-5 \right) \right)$
After taking out the common factor from the terms. Finally, we get two terms.
$\Rightarrow -1\left( 2x-7 \right)\left( 3x-5 \right)$
Now taking the first term and equating to 0, we get
$\Rightarrow \left( 2x-7 \right)=0$
$\Rightarrow 2x=7$
$\Rightarrow x=\dfrac{7}{2}$
Now taking the second term and equating to 0, we get
$\Rightarrow \left( 3x-5 \right)=0$
$\Rightarrow 3x=5$
$\Rightarrow x=\dfrac{5}{3}$
Therefore the factors for the equation $-6{{x}^{2}}+31x-35$ are $x=\dfrac{7}{2}$ and $x=\dfrac{5}{3}$.


Note: We solved the above equation by taking the common factors. But there are a total of five types of factorization of quadratic equations. Categorization is depending on the type of equation. The five types of the factorization are factorization by taking out the common factors, factorization by grouping the terms, factorization by making a perfect square, factorization by the difference of two squares, factoring the sum and difference of cube of two quantities.