Alright, let's factorize the given quadratic expressions step-by-step:
### a) Factorize [tex]\(x^2 + 3x - 4\)[/tex]
To factorize the quadratic expression [tex]\(x^2 + 3x - 4\)[/tex], we need to find two numbers that multiply to the constant term [tex]\(-4\)[/tex] and add up to the coefficient of the linear term, which is [tex]\(3\)[/tex].
1. Find the product and sum:
- Product = [tex]\(-4\)[/tex]
- Sum = [tex]\(3\)[/tex]
2. Identify two numbers that meet these conditions:
- The two numbers are [tex]\(4\)[/tex] and [tex]\(-1\)[/tex]:
- [tex]\(4 \times (-1) = -4\)[/tex]
- [tex]\(4 + (-1) = 3\)[/tex]
3. Write the quadratic expression as a product of two binomials:
- [tex]\((x + 4)(x - 1)\)[/tex]
So, [tex]\(x^2 + 3x - 4\)[/tex] can be factorized as:
[tex]\[ (x - 1)(x + 4) \][/tex]
### b) Factorize [tex]\(x^2 - 2x - 3\)[/tex]
To factorize the quadratic expression [tex]\(x^2 - 2x - 3\)[/tex], we need to find two numbers that multiply to the constant term [tex]\(-3\)[/tex] and add up to the coefficient of the linear term, which is [tex]\(-2\)[/tex].
1. Find the product and sum:
- Product = [tex]\(-3\)[/tex]
- Sum = [tex]\(-2\)[/tex]
2. Identify two numbers that meet these conditions:
- The two numbers are [tex]\(-3\)[/tex] and [tex]\(1\)[/tex]:
- [tex]\(-3 \times 1 = -3\)[/tex]
- [tex]\(-3 + 1 = -2\)[/tex]
3. Write the quadratic expression as a product of two binomials:
- [tex]\((x - 3)(x + 1)\)[/tex]
So, [tex]\(x^2 - 2x - 3\)[/tex] can be factorized as:
[tex]\[ (x - 3)(x + 1) \][/tex]
### c) Factorize [tex]\(x^2 + 2x - 8\)[/tex]
To factorize the quadratic expression [tex]\(x^2 + 2x - 8\)[/tex], we need to find two numbers that multiply to the constant term [tex]\(-8\)[/tex] and add up to the coefficient of the linear term, which is [tex]\(2\)[/tex].
1. Find the product and sum:
- Product = [tex]\(-8\)[/tex]
- Sum = [tex]\(2\)[/tex]
2. Identify two numbers that meet these conditions:
- The two numbers are [tex]\(4\)[/tex] and [tex]\(-2\)[/tex]:
- [tex]\(4 \times (-2) = -8\)[/tex]
- [tex]\(4 + (-2) = 2\)[/tex]
3. Write the quadratic expression as a product of two binomials:
- [tex]\((x + 4)(x - 2)\)[/tex]
So, [tex]\(x^2 + 2x - 8\)[/tex] can be factorized as:
[tex]\[ (x - 2)(x + 4) \][/tex]
### Summary
The factorized forms of the given expressions are:
a) [tex]\(x^2 + 3x - 4\)[/tex] is [tex]\((x - 1)(x + 4)\)[/tex].
b) [tex]\(x^2 - 2x - 3\)[/tex] is [tex]\((x - 3)(x + 1)\)[/tex].
c) [tex]\(x^2 + 2x - 8\)[/tex] is [tex]\((x - 2)(x + 4)\)[/tex].
