Question

How many real roots are in \[y = 2{x^2} - 7x - 15\]?

Answer 1

Hint: Here, we will use the concept of the factorization. Factorization is the process in which a number is written in the forms of its small factors which on multiplication give the original number. First, we will split the middle term of the given equation. Then we will form the factors by taking out the common terms in the equation. Then we will obtain the roots of the equation using the obtained factors.

Complete step by step solution:
The given equation is \[y = 2{x^2} - 7x - 15\].
First, we will form the factors of the quadratic equation by splitting the middle term into two parts such that its product will be equal to the product of the first term and the third term of the equation. Therefore, we get
\[ \Rightarrow y = 2{x^2} + 3x - 10x - 15\]
Now we will be taking \[x\] common from the first two terms and taking \[ - 5\] common from the last two terms. Therefore, the equation becomes
\[ \Rightarrow y = x\left( {2x + 3} \right) - 5\left( {2x + 3} \right)\]
We will take \[\left( {2x + 3} \right)\] common from the equation, we get
\[ \Rightarrow y = \left( {2x + 3} \right)\left( {x - 5} \right)\]
Now we will equate the factor to zero to get the value of \[x\].
\[\left( {2x + 3} \right)\left( {x - 5} \right) = 0\]
Using zero product property, we get
\[\begin{array}{l} \Rightarrow \left( {2x + 3} \right) = 0\\ \Rightarrow x = - \dfrac{3}{2}\end{array}\]
Or
 \[\begin{array}{l} \Rightarrow \left( {x - 5} \right) = 0\\ \Rightarrow x = 5\end{array}\]

Hence, there are only two roots of the given equation i.e. \[x = \dfrac{{ - 3}}{2}\] and \[5\].

Note:
The given equation is a quadratic equation. A quadratic equation is an equation that has the highest degree of 2 and has 2 solutions. Here, we can split the middle term according to the basic condition. The condition states that the middle term should be divided in such a way that its product must be equal to the product of the first and the last term of the equation. Factors are the smallest part of the number or equation which on multiplication will give us the actual number of equations.