Factorise
a.) [tex]x^2-18 x+56[/tex]
Answer (2)
Find two numbers that multiply to 56 and add up to -18: -4 and -14.
Rewrite the quadratic: x 2 − 18 x + 56 = x 2 − 4 x − 14 x + 56 .
Factor by grouping: x ( x − 4 ) − 14 ( x − 4 ) .
Factor out the common factor: ( x − 4 ) ( x − 14 ) . The final answer is ( x − 4 ) ( x − 14 ) .
Explanation
Understanding the problem We are asked to factorise the quadratic expression x 2 − 18 x + 56 . This means we need to find two binomials that, when multiplied together, give us the original quadratic expression.
Finding the right numbers To factorise x 2 − 18 x + 56 , we need to find two numbers that multiply to 56 (the constant term) and add up to -18 (the coefficient of the x term).
Identifying the correct pair Let's list the pairs of factors of 56:
(1, 56) (2, 28) (4, 14) (7, 8)
Since we need the factors to add up to -18, we consider the negative pairs:
(-1, -56) (-2, -28) (-4, -14) (-7, -8)
The pair (-4, -14) adds up to -18.
Rewriting the expression Now we can rewrite the quadratic expression using these two numbers:
x 2 − 18 x + 56 = x 2 − 4 x − 14 x + 56
Factoring by grouping Next, we factor by grouping:
x 2 − 4 x − 14 x + 56 = x ( x − 4 ) − 14 ( x − 4 )
Final factorization Finally, we factor out the common factor ( x − 4 ) :
x ( x − 4 ) − 14 ( x − 4 ) = ( x − 4 ) ( x − 14 )
The answer So, the factorised form of x 2 − 18 x + 56 is ( x − 4 ) ( x − 14 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, suppose you are designing a rectangular garden and you know the area must be x 2 − 18 x + 56 square feet. By factoring this expression into ( x − 4 ) ( x − 14 ) , you determine that the dimensions of the garden could be ( x − 4 ) feet and ( x − 14 ) feet. This allows you to plan the layout of your garden based on the desired area.
The expression x 2 − 18 x + 56 can be factored as ( x − 4 ) ( x − 14 ) by finding two numbers that multiply to 56 and add to -18. These numbers are -4 and -14. By rewriting the quadratic and applying the grouping method, we arrive at the final factorised form.
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