Question

How do you factor $ 4{x^2} - 16x + 15 $ ?

Answer 1

Hint: We are given a quadratic algebraic expression of the form $ a{x^2} + bx + c $ it is a quadratic expression. And we have to find its factor. We will find the factor by splitting the middle term for the equation $ a{x^2} + bx + c $ . Here the first term is $ a{x^2} $ its coefficient is $ a $ . The middle term is $ - bx $ . Coefficient is $ b $ . The last term is constant i.e. $ + c $ . As there are + signs between second term and constant term. So we will split the middle term in the form of a sum of two numbers with minus sign.

Complete step-by-step solution:
Step1: We are given an expression $ 4{x^2} - 16x + 15 $
Now we will try to factor out the term by splitting the middle term:
The first term is, $ 4{x^2} $ its coefficient is $ 4 $ . The middle term is, $ - 16x $ its coefficient is $ 16 $ . The last term, ‘the constant’, is $ + 15 $ . We will multiply the coefficient of the first term by the constant
 $ \Rightarrow 4 \times 15 = 60 $
Step2: Now we will find the two factors of $ 60 $ whose sum equals the coefficient of the middle term, which is $ - 16 $ . On doing the factors we get the factors i.e. $ 2,2,3,5 $
We can arrange $ 60 $ as
 $ \Rightarrow 60 = 2 \times 3 \times 2 \times 5 $ .
 Now we will arrange these terms to get two numbers whose sum will be equal to $ 16 $ so two numbers we get are $ - 10\& - 6 $ now we will rewrite the polynomial by splitting the middle term we get:
 $ \Rightarrow 4{x^2} - 10x - 6x + 15 $
Step3: Add up the first two terms, pulling out the like factors:
 $ \Rightarrow 2x\left( {2x - 5} \right) $
Add up the last two terms, pulling out common factors:
 $ \Rightarrow - 3(2x - 5) $
Step4: Add up the four terms of step3 we will get:
 $ \Rightarrow \left( {2x - 5} \right)\left( {2x - 3} \right) $
Which is the desired factorization.

Hence the desired factor is $ \left( {2x - 3} \right)\left( {2x - 5} \right) $

Note: In such types of questions if we are given a quadratic equation then it can be easily solved by splitting the middle term. If the equation is a simple form but if the equation contain big coefficients or any rational coefficients then use the quadratic formula i.e. $ \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ by using this we can solve any complicated equation. But for small simple equation then we will use the splitting the middle term method for this also there are two types of equation form for $ a{x^2} + bx + c $ we will split the middle term in form of addition and for $ a{x^2} + bx - c $ . We will split the middle term in form subtraction. Remember this while solving by splitting the middle term.