Fully factorise the quadratic expression $7 b-b^2-10$

Rewrite the expression in standard quadratic form: − b 2 + 7 b − 10 .
Factor out − 1 : − ( b 2 − 7 b + 10 ) .
Factor the quadratic inside the parentheses: − ( b − 2 ) ( b − 5 ) .
Distribute the negative sign to one of the factors: ( 2 − b ) ( b − 5 ) .
The fully factorised form is ( 2 − b ) ( b − 5 ) ​ .

Explanation

Understanding the problem We are asked to fully factorise the quadratic expression 7 b − b 2 − 10 . This means we need to rewrite the expression as a product of two linear expressions in terms of b .

Rewriting the expression First, let's rewrite the expression in the standard quadratic form a x 2 + b x + c , where x is our variable. In this case, our variable is b , so we have:


7 b − b 2 − 10 = − b 2 + 7 b − 10

Factoring out -1 Next, we factor out a − 1 from the expression to make the leading coefficient positive. This gives us:

− ( b 2 − 7 b + 10 )

Factoring the quadratic Now, we need to factorise the quadratic expression inside the parentheses, b 2 − 7 b + 10 . We are looking for two numbers that multiply to 10 and add up to -7. These numbers are -2 and -5, since ( − 2 ) × ( − 5 ) = 10 and ( − 2 ) + ( − 5 ) = − 7 . Therefore, we can write the quadratic expression as:

( b − 2 ) ( b − 5 )

Substituting back Substituting this back into the expression with the − 1 factored out, we have:

− ( b − 2 ) ( b − 5 )

Distributing the negative sign Finally, we can distribute the negative sign into one of the factors. For example, distributing it into the first factor, we get:

( − b + 2 ) ( b − 5 ) = ( 2 − b ) ( b − 5 )
Alternatively, we can distribute it into the second factor to get:
( b − 2 ) ( − b + 5 ) = ( b − 2 ) ( 5 − b )
Both of these expressions are equivalent.

Final factorised form Therefore, the fully factorised form of the quadratic expression 7 b − b 2 − 10 is ( 2 − b ) ( b − 5 ) or ( 5 − b ) ( b − 2 ) .

Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, imagine you are designing a rectangular garden and you know the area can be expressed as 7 b − b 2 − 10 , where b is related to the dimensions of the garden. By factoring this expression into ( 2 − b ) ( b − 5 ) , you can determine the possible values for the length and width of the garden. This helps you plan the layout and optimize the use of space in your garden design.

Answered by GinnyAnswer | 2025-07-03

To factor the quadratic expression 7 b − b 2 − 10 , we rewrite it in standard form, factor out − 1 , and then factor the quadratic using two numbers that multiply to 10 and sum to -7. This leads to the factorized form of ( 2 − b ) ( b − 5 ) or ( 5 − b ) ( b − 2 ) . Both forms are correct and equivalent.
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Answered by Anonymous | 2025-07-04