Factor the following expression: [tex]$2 x^2-11 x+9$[/tex]
Answer (2)
Find two numbers that multiply to 2 ⋅ 9 = 18 and add up to − 11 , which are − 2 and − 9 .
Rewrite the middle term: 2 x 2 − 11 x + 9 = 2 x 2 − 2 x − 9 x + 9 .
Factor by grouping: 2 x ( x − 1 ) − 9 ( x − 1 ) .
Factor out the common factor: ( 2 x − 9 ) ( x − 1 ) . The factored expression is ( 2 x − 9 ) ( x − 1 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression 2 x 2 − 11 x + 9 . Factoring a quadratic means rewriting it as a product of two binomials. We look for two binomials of the form ( a x + b ) ( c x + d ) such that when multiplied, they give us the original quadratic.
Finding the Right Numbers To factor the quadratic 2 x 2 − 11 x + 9 , we need to find two numbers that multiply to 2 × 9 = 18 and add up to − 11 . These two numbers are − 2 and − 9 .
Rewriting the Middle Term Now we rewrite the middle term of the quadratic using these two numbers: 2 x 2 − 11 x + 9 = 2 x 2 − 2 x − 9 x + 9
Factoring by Grouping Next, we factor by grouping. We group the first two terms and the last two terms: ( 2 x 2 − 2 x ) + ( − 9 x + 9 ) Now, we factor out the greatest common factor from each group: 2 x ( x − 1 ) − 9 ( x − 1 )
Factoring out the Common Factor We notice that ( x − 1 ) is a common factor in both terms. We factor out ( x − 1 ) : ( 2 x − 9 ) ( x − 1 ) So, the factored form of the quadratic expression is ( 2 x − 9 ) ( x − 1 ) .
Final Answer Therefore, the factored form of 2 x 2 − 11 x + 9 is ( 2 x − 9 ) ( x − 1 ) .
Examples
Factoring quadratic expressions is useful in many real-world situations. For example, suppose you are designing a rectangular garden and you know the area can be represented by the expression 2 x 2 − 11 x + 9 . By factoring this expression into ( 2 x − 9 ) ( x − 1 ) , you can determine the possible dimensions of the garden in terms of x . This allows you to plan the layout and optimize the use of space based on the value of x .
To factor the expression 2 x 2 − 11 x + 9 , we rewrite it as ( 2 x − 9 ) ( x − 1 ) after finding two numbers that multiply to 18 and add to -11. This involves grouping and factoring by recognizing common factors. The final factored form is ( 2 x − 9 ) ( x − 1 ) .
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