Answer :
To determine which of the given options are solutions to the equation [tex]\(4x^2 - 12x + 9 = 5\)[/tex], we need to follow a few steps:
1. Rewriting the Equation:
Start by rewriting the equation in standard quadratic form:
[tex]\[ 4x^2 - 12x + 9 - 5 = 0 \implies 4x^2 - 12x + 4 = 0 \][/tex]
2. Simplifying the Quadratic Equation:
Simplify the quadratic equation by dividing by 4:
[tex]\[ x^2 - 3x + 1 = 0 \][/tex]
3. Solving the Quadratic Equation:
To solve the quadratic equation [tex]\(x^2 - 3x + 1 = 0\)[/tex], we can use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = 1\)[/tex].
[tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{9 - 4}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{5}}{2} \][/tex]
Therefore, the solutions are:
[tex]\[ x_1 = \frac{3 + \sqrt{5}}{2}, \quad x_2 = \frac{3 - \sqrt{5}}{2} \][/tex]
4. Matching Solutions with Given Options:
Now let's match each of the given options with the solutions [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]:
- Option A: [tex]\(x = \frac{\sqrt{5} + 3}{2}\)[/tex]
[tex]\[ \frac{3 + \sqrt{5}}{2} \quad \textit{matches} \quad x_1 = \frac{3 + \sqrt{5}}{2} \][/tex]
Therefore, Option A is a solution.
- Option B: [tex]\(x = \sqrt{4} - 3\)[/tex]
[tex]\[ \sqrt{4} - 3 = 2 - 3 = -1 \quad (\textit{This does not match} \quad x_1 \textit{ or } x_2) \][/tex]
Therefore, Option B is not a solution.
- Option C: [tex]\(x = -\sqrt{4} - 3\)[/tex]
[tex]\[ -\sqrt{4} - 3 = -2 - 3 = -5 \quad (\textit{This does not match} \quad x_1 \textit{ or } x_2) \][/tex]
Therefore, Option C is not a solution.
- Option D: [tex]\(x = -\sqrt{5} + \frac{3}{2}\)[/tex]
[tex]\[ -\sqrt{5} + \frac{3}{2} \quad (\textit{This does not match} \quad x_1 \textit{ or } x_2) \][/tex]
Therefore, Option D is not a solution.
- Option E: [tex]\(x = \sqrt{5} + \frac{3}{2}\)[/tex]
[tex]\[ \sqrt{5} + \frac{3}{2} \quad (\textit{This does not match} \quad x_1 \textit{ or } x_2) \][/tex]
Therefore, Option E is not a solution.
- Option F: [tex]\(x = \frac{-\sqrt{5} + 3}{2}\)[/tex]
[tex]\[ \frac{3 - \sqrt{5}}{2} \quad \textit{matches} \quad x_2 = \frac{3 - \sqrt{5}}{2} \][/tex]
Therefore, Option F is a solution.
In summary, the following are the solutions to the equation [tex]\(4x^2 - 12x + 9 = 5\)[/tex]:
[tex]\[A. \quad x = \frac{\sqrt{5} + 3}{2} \][/tex]
[tex]\[F. \quad x = \frac{-\sqrt{5} + 3}{2} \][/tex]
1. Rewriting the Equation:
Start by rewriting the equation in standard quadratic form:
[tex]\[ 4x^2 - 12x + 9 - 5 = 0 \implies 4x^2 - 12x + 4 = 0 \][/tex]
2. Simplifying the Quadratic Equation:
Simplify the quadratic equation by dividing by 4:
[tex]\[ x^2 - 3x + 1 = 0 \][/tex]
3. Solving the Quadratic Equation:
To solve the quadratic equation [tex]\(x^2 - 3x + 1 = 0\)[/tex], we can use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = 1\)[/tex].
[tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{9 - 4}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{5}}{2} \][/tex]
Therefore, the solutions are:
[tex]\[ x_1 = \frac{3 + \sqrt{5}}{2}, \quad x_2 = \frac{3 - \sqrt{5}}{2} \][/tex]
4. Matching Solutions with Given Options:
Now let's match each of the given options with the solutions [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]:
- Option A: [tex]\(x = \frac{\sqrt{5} + 3}{2}\)[/tex]
[tex]\[ \frac{3 + \sqrt{5}}{2} \quad \textit{matches} \quad x_1 = \frac{3 + \sqrt{5}}{2} \][/tex]
Therefore, Option A is a solution.
- Option B: [tex]\(x = \sqrt{4} - 3\)[/tex]
[tex]\[ \sqrt{4} - 3 = 2 - 3 = -1 \quad (\textit{This does not match} \quad x_1 \textit{ or } x_2) \][/tex]
Therefore, Option B is not a solution.
- Option C: [tex]\(x = -\sqrt{4} - 3\)[/tex]
[tex]\[ -\sqrt{4} - 3 = -2 - 3 = -5 \quad (\textit{This does not match} \quad x_1 \textit{ or } x_2) \][/tex]
Therefore, Option C is not a solution.
- Option D: [tex]\(x = -\sqrt{5} + \frac{3}{2}\)[/tex]
[tex]\[ -\sqrt{5} + \frac{3}{2} \quad (\textit{This does not match} \quad x_1 \textit{ or } x_2) \][/tex]
Therefore, Option D is not a solution.
- Option E: [tex]\(x = \sqrt{5} + \frac{3}{2}\)[/tex]
[tex]\[ \sqrt{5} + \frac{3}{2} \quad (\textit{This does not match} \quad x_1 \textit{ or } x_2) \][/tex]
Therefore, Option E is not a solution.
- Option F: [tex]\(x = \frac{-\sqrt{5} + 3}{2}\)[/tex]
[tex]\[ \frac{3 - \sqrt{5}}{2} \quad \textit{matches} \quad x_2 = \frac{3 - \sqrt{5}}{2} \][/tex]
Therefore, Option F is a solution.
In summary, the following are the solutions to the equation [tex]\(4x^2 - 12x + 9 = 5\)[/tex]:
[tex]\[A. \quad x = \frac{\sqrt{5} + 3}{2} \][/tex]
[tex]\[F. \quad x = \frac{-\sqrt{5} + 3}{2} \][/tex]
