Question

How do you solve $3x^2-2x-5=0$ ?

Answer 1

Hint: In this question, we need to solve the equation $3x^2-2x-5=0$ . For splitting the middle term into two factors, we will determine the factors that multiply to give $ac$ i.e., $3\times -5=-15$ , and add to give $b$ i.e., $-2$ which is called a sum-product pattern. Then, factor the first two and last two terms separately. If we have done this correctly, then two new terms will have a clearly visible common factor. Finally, we will equate the factors to $0$ and determine the value of $x$ .

Complete step-by-step answer:
Now, we need to solve $3x^2-2x-5=0$ .
First, let us determine the factors of the given equation.
According to the rule to factorize,
Product= $x^2$ coefficient $\times$ constant
And, sum= $x$ coefficient
Thus, we will find two numbers that multiply to give $ac$ i.e., $3\times -5=-15$ and add to give $b$ i.e., $-2$ ,
Here, the product is negative. So, we can say that one of the factors is negative, and then the other is positive.
Now, let’s consider the possible factors and their sum.
$$\begin{gathered} 15\times -1=-15;15+\left(-1\right)=14 \\ 5\times -3=-15;5+\left(-3\right)=2 \\ 3\times -5=-15;3+\left(-5\right)=-2 \\ 1\times -15=-15;1+\left(-15\right)=-14 \end{gathered}$$
From this it is clear that the factors are $3$ and $-5$ .
Now, by rewriting the middle term with those factors, we have,
 $3x^2+3x-5x-5=0$
 $\left(3x^2+3x\right)-\left(5x+5\right)=0$
Factor out the greatest common factor from each group,
 $3x\left(x+1\right)-5\left(x+1\right)=0$
Factor the polynomial by factoring out the greatest common factor, $x+1$ ,
 $\Rightarrow \left(3x-5\right)\left(x+1\right)=0$
Now, equate the factors separately to determine the value of $x$ .
Therefore, $\left(3x-5\right)=0$ and $\left(x+1\right)=0$
Hence, $x={\displaystyle \frac{5}{3}}$ and $x=-1$ .
So, the correct answer is “ $x={\displaystyle \frac{5}{3}}$ and $x=-1$”.

Note: In this question it is important to note that this factorization method works for all quadratic equations. The standard form of the quadratic equation is $a{x}^2+bx+c=0$ . It is called factoring because we find the factors. A factor is something we multiply by. There is no simple method of factorizing a quadratic expression, but with a little practice it becomes easier. Finally, equating the equation to $0$ is common in all quadratic equations because we need to determine the value of the given unknown variable.