Question

How do you factor the expression $3{x^2} + 15x - 42 = 0$ ?

Answer 1

Hint: A polynomial is an algorithmic function that contains the numerical values as the coefficient of the unknown variable (denoted by alphabets) raised to some power. The highest exponent in a polynomial equation is known as the degree of the polynomial. A polynomial of degree two is called a quadratic polynomial so the given equation is quadratic.

Complete step by step answer:
We know that a polynomial equation has exactly as many roots as its degree so the given equation will have two roots as it is a quadratic equation. The roots of an equation can be found out using various methods like factorization, completing the square, graph, quadratic formula, etc. but we are told to solve the given equation by factorization, so we solve the question by making the factors of the given equation.
We have $3{x^2} + 15x - 42 = 0$
$
   \Rightarrow 3({x^2} + 5x - 14) = 0 \\
   \Rightarrow {x^2} + 5x - 14 = 0 \\
 $
Factorizing the above equation, we get –
$
  {x^2} + 7x - 2x - 14 = 0 \\
   \Rightarrow x(x + 7) - 2(x + 7) = 0 \\
   \Rightarrow (x + 7)(x - 2) = 0 \\
   \Rightarrow x = - 7,\,x = 2 \\
 $
Hence, on factorizing the expression $3{x^2} + 15x - 42 = 0$ , the factors obtained are $x + 7 = 0$ and $x - 2 = 0$ .

Note: The standard form of a quadratic equation is $a{x^2} + bx + c = 0$, where a is the coefficient of ${x^2}$ , b is the coefficient of $x$ and c is a constant value. For solving an equation by factorization, the condition to form factors is that we have to express b as a sum of two numbers such that their product is equal to the product of $a$ and $c$ that is $a \times c = {b_1} \times {b_2}$ . -14 can be written as a product of 2 and -7 or 7 and -2 but on adding 7 and -2, we get -5, This is how we factorize a quadratic equation.