Question

How do you factor the trinomial \[{x^2} - 13x - 14\]?

Answer 1

Hint: In these type polynomials, we will solve by using splitting middle term, for a polynomial of the form \[a{x^2} + bx + c = 0\], rewrite the middle term as a sum of two terms whose product is \[a \cdot c = 1 \cdot - 14 = - 14\] and whose sum is \[b = - 13\], when you solve the expression we will get the required values.

Complete step-by-step solution:
A trinomial is an algebraic equation which will have three terms. Quadratic equations are the equations that are often called second degree. It means that it consists at least one term which is squared, the general form of quadratic equation is \[a{x^2} + bx + c = 0\], where "\[a\]", "\[b\]", and "\[c\]"are numerical coefficients or constant, and the value of\[x\]is unknown. And one fundamental rule is that the value of \[a\], the first constant cannot be zero in a quadratic equation
Now given equation is \[{y^2} - 6y + 8 = 0\],
This can be factored by splitting the middle term, now rewrite the middle term as a sum of two terms, we will get two terms. Now sum of two terms whose product is \[a \cdot c = 1 \cdot - 14 = - 14\] and whose sum is \[b = - 13\],
Now we will find the numbers which satisfy the given data, i.e., sum of the numbers should be equal to -13 and their product should be equal to -14,
So, now rewrite the -13 as -14 and 1, whose product will be equal to -14 and their sum is -13.
Now using distributive property, we get
\[ \Rightarrow {x^2} - 14x + 1x - 14\],
By grouping the first two terms and last two terms, we get,
\[ \Rightarrow \left( {{x^2} - 14x} \right) + \left( {x - 14} \right)\],
Now factor out the highest common factor, we get
\[ \Rightarrow x\left( {x - 14} \right) + 1\left( {x - 14} \right)\],
Now taking common term in both, we get,
\[ \Rightarrow \left( {x - 14} \right)\left( {x + 1} \right)\],
The factorising terms of the given polynomial is \[\left( {x - 14} \right)\left( {x + 1} \right)\].

\[\therefore \] The factorising terms when the given polynomial \[{x^2} - 13x - 14\] is factorised will be equal to \[\left( {x - 14} \right)\left( {x + 1} \right)\].

Note: We have several options for factoring when you are solving the polynomial equations, one method is method of using Quadratic equation formula and it is a method of solving quadratic equations, but we should keep in mind that we can also solve the equation using completely the square, and we can cross check the values of \[x\] by using the above formula. Also we should always convert the coefficient of \[{x^2} = 1\], to easily solve the equation by this method, and there are other methods to solve such kind of solutions, other method used to solve the quadratic equation is by factoring method, in this method we should obtain the solution factorising quadratic equation terms. In these types of questions, we can solve by using quadratic formula i.e.,\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].