Answer :
Let's tackle each equation one by one, placing them into factored form and solving for the [tex]\( x \)[/tex]-intercepts.
### a) [tex]\( f(x) = 2x^2 + 7x + 5 \)[/tex]
1. Factor the quadratic equation:
[tex]\[ f(x) = (x + 1)(2x + 5) \][/tex]
2. Solve for the [tex]\( x \)[/tex]-intercepts by setting each factor equal to zero:
[tex]\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \][/tex]
[tex]\[ 2x + 5 = 0 \quad \Rightarrow \quad x = -\frac{5}{2} \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = -1 \)[/tex] and [tex]\( x = -\frac{5}{2} \)[/tex].
### b) [tex]\( f(x) = 2x^2 + 7x + 3 \)[/tex]
1. Factor the quadratic equation:
[tex]\[ f(x) = (x + 3)(2x + 1) \][/tex]
2. Solve for the [tex]\( x \)[/tex]-intercepts by setting each factor equal to zero:
[tex]\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \][/tex]
[tex]\[ 2x + 1 = 0 \quad \Rightarrow \quad x = -\frac{1}{2} \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = -3 \)[/tex] and [tex]\( x = -\frac{1}{2} \)[/tex].
### c) [tex]\( f(x) = 6x^2 - 13x + 6 \)[/tex]
1. Factor the quadratic equation:
[tex]\[ f(x) = (2x - 3)(3x - 2) \][/tex]
2. Solve for the [tex]\( x \)[/tex]-intercepts by setting each factor equal to zero:
[tex]\[ 2x - 3 = 0 \quad \Rightarrow \quad x = \frac{3}{2} \][/tex]
[tex]\[ 3x - 2 = 0 \quad \Rightarrow \quad x = \frac{2}{3} \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = \frac{3}{2} \)[/tex] and [tex]\( x = \frac{2}{3} \)[/tex].
### d) [tex]\( f(x) = 8x^2 - 14x + 5 \)[/tex]
1. Factor the quadratic equation:
[tex]\[ f(x) = (2x - 1)(4x - 5) \][/tex]
2. Solve for the [tex]\( x \)[/tex]-intercepts by setting each factor equal to zero:
[tex]\[ 2x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{2} \][/tex]
[tex]\[ 4x - 5 = 0 \quad \Rightarrow \quad x = \frac{5}{4} \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = \frac{1}{2} \)[/tex] and [tex]\( x = \frac{5}{4} \)[/tex].
### e) [tex]\( f(x) = -8x^2 - 2x + 1 \)[/tex]
1. Factor the quadratic equation:
[tex]\[ f(x) = -(2x + 1)(4x - 1) \][/tex]
2. Solve for the [tex]\( x \)[/tex]-intercepts by setting each factor equal to zero:
[tex]\[ 2x + 1 = 0 \quad \Rightarrow \quad x = -\frac{1}{2} \][/tex]
[tex]\[ 4x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{4} \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = -\frac{1}{2} \)[/tex] and [tex]\( x = \frac{1}{4} \)[/tex].
In summary, the factored forms and [tex]\( x \)[/tex]-intercepts are as follows:
1. [tex]\( f(x) = 2x^2 + 7x + 5 \)[/tex]
- Factored Form: [tex]\((x + 1)(2x + 5)\)[/tex]
- Interceptions: [tex]\( x = -1, -\frac{5}{2} \)[/tex]
2. [tex]\( f(x) = 2x^2 + 7x + 3 \)[/tex]
- Factored Form: [tex]\((x + 3)(2x + 1)\)[/tex]
- Interceptions: [tex]\( x = -3, -\frac{1}{2} \)[/tex]
3. [tex]\( f(x) = 6x^2 - 13x + 6 \)[/tex]
- Factored Form: [tex]\((2x - 3)(3x - 2)\)[/tex]
- Interceptions: [tex]\( x = \frac{3}{2}, \frac{2}{3} \)[/tex]
4. [tex]\( f(x) = 8x^2 - 14x + 5 \)[/tex]
- Factored Form: [tex]\((2x - 1)(4x - 5)\)[/tex]
- Interceptions: [tex]\( x = \frac{1}{2}, \frac{5}{4} \)[/tex]
5. [tex]\( f(x) = -8x^2 - 2x + 1 \)[/tex]
- Factored Form: [tex]\(-(2x + 1)(4x - 1)\)[/tex]
- Interceptions: [tex]\( x = -\frac{1}{2}, \frac{1}{4} \)[/tex]
These steps provide a clear pathway to factorize these quadratic equations and determine their [tex]\(x\)[/tex]-intercepts.
### a) [tex]\( f(x) = 2x^2 + 7x + 5 \)[/tex]
1. Factor the quadratic equation:
[tex]\[ f(x) = (x + 1)(2x + 5) \][/tex]
2. Solve for the [tex]\( x \)[/tex]-intercepts by setting each factor equal to zero:
[tex]\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \][/tex]
[tex]\[ 2x + 5 = 0 \quad \Rightarrow \quad x = -\frac{5}{2} \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = -1 \)[/tex] and [tex]\( x = -\frac{5}{2} \)[/tex].
### b) [tex]\( f(x) = 2x^2 + 7x + 3 \)[/tex]
1. Factor the quadratic equation:
[tex]\[ f(x) = (x + 3)(2x + 1) \][/tex]
2. Solve for the [tex]\( x \)[/tex]-intercepts by setting each factor equal to zero:
[tex]\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \][/tex]
[tex]\[ 2x + 1 = 0 \quad \Rightarrow \quad x = -\frac{1}{2} \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = -3 \)[/tex] and [tex]\( x = -\frac{1}{2} \)[/tex].
### c) [tex]\( f(x) = 6x^2 - 13x + 6 \)[/tex]
1. Factor the quadratic equation:
[tex]\[ f(x) = (2x - 3)(3x - 2) \][/tex]
2. Solve for the [tex]\( x \)[/tex]-intercepts by setting each factor equal to zero:
[tex]\[ 2x - 3 = 0 \quad \Rightarrow \quad x = \frac{3}{2} \][/tex]
[tex]\[ 3x - 2 = 0 \quad \Rightarrow \quad x = \frac{2}{3} \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = \frac{3}{2} \)[/tex] and [tex]\( x = \frac{2}{3} \)[/tex].
### d) [tex]\( f(x) = 8x^2 - 14x + 5 \)[/tex]
1. Factor the quadratic equation:
[tex]\[ f(x) = (2x - 1)(4x - 5) \][/tex]
2. Solve for the [tex]\( x \)[/tex]-intercepts by setting each factor equal to zero:
[tex]\[ 2x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{2} \][/tex]
[tex]\[ 4x - 5 = 0 \quad \Rightarrow \quad x = \frac{5}{4} \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = \frac{1}{2} \)[/tex] and [tex]\( x = \frac{5}{4} \)[/tex].
### e) [tex]\( f(x) = -8x^2 - 2x + 1 \)[/tex]
1. Factor the quadratic equation:
[tex]\[ f(x) = -(2x + 1)(4x - 1) \][/tex]
2. Solve for the [tex]\( x \)[/tex]-intercepts by setting each factor equal to zero:
[tex]\[ 2x + 1 = 0 \quad \Rightarrow \quad x = -\frac{1}{2} \][/tex]
[tex]\[ 4x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{4} \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = -\frac{1}{2} \)[/tex] and [tex]\( x = \frac{1}{4} \)[/tex].
In summary, the factored forms and [tex]\( x \)[/tex]-intercepts are as follows:
1. [tex]\( f(x) = 2x^2 + 7x + 5 \)[/tex]
- Factored Form: [tex]\((x + 1)(2x + 5)\)[/tex]
- Interceptions: [tex]\( x = -1, -\frac{5}{2} \)[/tex]
2. [tex]\( f(x) = 2x^2 + 7x + 3 \)[/tex]
- Factored Form: [tex]\((x + 3)(2x + 1)\)[/tex]
- Interceptions: [tex]\( x = -3, -\frac{1}{2} \)[/tex]
3. [tex]\( f(x) = 6x^2 - 13x + 6 \)[/tex]
- Factored Form: [tex]\((2x - 3)(3x - 2)\)[/tex]
- Interceptions: [tex]\( x = \frac{3}{2}, \frac{2}{3} \)[/tex]
4. [tex]\( f(x) = 8x^2 - 14x + 5 \)[/tex]
- Factored Form: [tex]\((2x - 1)(4x - 5)\)[/tex]
- Interceptions: [tex]\( x = \frac{1}{2}, \frac{5}{4} \)[/tex]
5. [tex]\( f(x) = -8x^2 - 2x + 1 \)[/tex]
- Factored Form: [tex]\(-(2x + 1)(4x - 1)\)[/tex]
- Interceptions: [tex]\( x = -\frac{1}{2}, \frac{1}{4} \)[/tex]
These steps provide a clear pathway to factorize these quadratic equations and determine their [tex]\(x\)[/tex]-intercepts.
