Answer :
To factorize the quadratic expression [tex]\(8x^2 - 18x - 35\)[/tex], we need to find two binomials that multiply to give the original expression. Let's go through the process step by step:
1. Identify the quadratic expression: We have [tex]\(8x^2 - 18x - 35\)[/tex].
2. Determine the coefficients:
- The coefficient of [tex]\(x^2\)[/tex] (quadratic term) is [tex]\(a = 8\)[/tex].
- The coefficient of [tex]\(x\)[/tex] (linear term) is [tex]\(b = -18\)[/tex].
- The constant term is [tex]\(c = -35\)[/tex].
3. Factorization process involves finding two numbers that multiply to the product of [tex]\(a\)[/tex] and [tex]\(c\)[/tex] (i.e., [tex]\(8 \times -35 = -280\)[/tex]) and also add up to [tex]\(b\)[/tex] (i.e., [tex]\(-18\)[/tex]).
4. Finding the numbers:
- We need to find two numbers whose product is [tex]\(-280\)[/tex] and whose sum is [tex]\(-18\)[/tex].
- These numbers are [tex]\(-28\)[/tex] and [tex]\(10\)[/tex].
5. Rewrite the middle term using the two numbers:
- Replace [tex]\(-18x\)[/tex] with [tex]\(-28x + 10x\)[/tex].
- The expression now becomes: [tex]\(8x^2 - 28x + 10x - 35\)[/tex].
6. Group the terms:
- Group the first two terms and the last two terms separately: [tex]\((8x^2 - 28x) + (10x - 35)\)[/tex].
7. Factor by grouping:
- Factor out the greatest common factor from each group.
- For [tex]\(8x^2 - 28x\)[/tex], factor out [tex]\(4x\)[/tex]: [tex]\(4x(2x - 7)\)[/tex].
- For [tex]\(10x - 35\)[/tex], factor out [tex]\(5\)[/tex]: [tex]\(5(2x - 7)\)[/tex].
8. Combine the factors:
- Notice that [tex]\(2x - 7\)[/tex] is a common factor.
- Factor out [tex]\(2x - 7\)[/tex], resulting in:
[tex]\((2x - 7)(4x + 5)\)[/tex].
So, the factorized form of [tex]\(8x^2 - 18x - 35\)[/tex] is [tex]\((2x - 7)(4x + 5)\)[/tex]. This completes the factorization process.
1. Identify the quadratic expression: We have [tex]\(8x^2 - 18x - 35\)[/tex].
2. Determine the coefficients:
- The coefficient of [tex]\(x^2\)[/tex] (quadratic term) is [tex]\(a = 8\)[/tex].
- The coefficient of [tex]\(x\)[/tex] (linear term) is [tex]\(b = -18\)[/tex].
- The constant term is [tex]\(c = -35\)[/tex].
3. Factorization process involves finding two numbers that multiply to the product of [tex]\(a\)[/tex] and [tex]\(c\)[/tex] (i.e., [tex]\(8 \times -35 = -280\)[/tex]) and also add up to [tex]\(b\)[/tex] (i.e., [tex]\(-18\)[/tex]).
4. Finding the numbers:
- We need to find two numbers whose product is [tex]\(-280\)[/tex] and whose sum is [tex]\(-18\)[/tex].
- These numbers are [tex]\(-28\)[/tex] and [tex]\(10\)[/tex].
5. Rewrite the middle term using the two numbers:
- Replace [tex]\(-18x\)[/tex] with [tex]\(-28x + 10x\)[/tex].
- The expression now becomes: [tex]\(8x^2 - 28x + 10x - 35\)[/tex].
6. Group the terms:
- Group the first two terms and the last two terms separately: [tex]\((8x^2 - 28x) + (10x - 35)\)[/tex].
7. Factor by grouping:
- Factor out the greatest common factor from each group.
- For [tex]\(8x^2 - 28x\)[/tex], factor out [tex]\(4x\)[/tex]: [tex]\(4x(2x - 7)\)[/tex].
- For [tex]\(10x - 35\)[/tex], factor out [tex]\(5\)[/tex]: [tex]\(5(2x - 7)\)[/tex].
8. Combine the factors:
- Notice that [tex]\(2x - 7\)[/tex] is a common factor.
- Factor out [tex]\(2x - 7\)[/tex], resulting in:
[tex]\((2x - 7)(4x + 5)\)[/tex].
So, the factorized form of [tex]\(8x^2 - 18x - 35\)[/tex] is [tex]\((2x - 7)(4x + 5)\)[/tex]. This completes the factorization process.
