Answer :
Sure, let's work on solving the given equations and simplifying the given expressions step-by-step.
### 1. Solve the equation [tex]\(11x^2 = 8x + 9\)[/tex]
First, let's rearrange the equation to standard quadratic form:
[tex]\[ 11x^2 - 8x - 9 = 0 \][/tex]
We use the quadratic formula to find the roots of the equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\(a = 11\)[/tex], [tex]\(b = -8\)[/tex], and [tex]\(c = -9\)[/tex]. Plugging these values into the quadratic formula gives:
[tex]\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(11)(-9)}}{2(11)} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{64 + 396}}{22} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{460}}{22} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{115 \times 4}}{22} \][/tex]
[tex]\[ x = \frac{8 \pm 2\sqrt{115}}{22} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{115}}{11} \][/tex]
So, the solutions are:
[tex]\[ x = \frac{4 - \sqrt{115}}{11} \quad \text{and} \quad x = \frac{4 + \sqrt{115}}{11} \][/tex]
### 2. Solve the equation [tex]\(2y^2 = 20y + 100\)[/tex]
First, let's rearrange the equation to standard quadratic form:
[tex]\[ 2y^2 - 20y - 100 = 0 \][/tex]
Divide every term by 2:
[tex]\[ y^2 - 10y - 50 = 0 \][/tex]
Using the quadratic formula:
[tex]\[ y = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(-50)}}{2(1)} \][/tex]
[tex]\[ y = \frac{10 \pm \sqrt{100 + 200}}{2} \][/tex]
[tex]\[ y = \frac{10 \pm \sqrt{300}}{2} \][/tex]
[tex]\[ y = \frac{10 \pm 10\sqrt{3}}{2} \][/tex]
[tex]\[ y = 5 \pm 5\sqrt{3} \][/tex]
So, the solutions are:
[tex]\[ y = 5 - 5\sqrt{3} \quad \text{and} \quad y = 5 + 5\sqrt{3} \][/tex]
### 3. Simplify the expression [tex]\(4x^2 + 16xy + 16y^2\)[/tex]
The given expression can be rewritten as:
[tex]\[ 4(x^2 + 4xy + 4y^2) \][/tex]
Notice that inside the parenthesis is a perfect square trinomial:
[tex]\[ x^2 + 4xy + 4y^2 = (x + 2y)^2 \][/tex]
So, the expression simplifies to:
[tex]\[ 4(x + 2y)^2 \][/tex]
### 4. Simplify the expression [tex]\(16a^2 - 240x + 4b^2\)[/tex]
The expression [tex]\(16a^2 - 240x + 4b^2\)[/tex] is already in its simplified form. There is no further simplification since there are no common factors or like terms that can be combined.
### 5. Simplify the expression [tex]\(16m^2 + 48m + 336\)[/tex]
This expression is already in its expanded form. However, we can factor out the common factor of 16:
[tex]\[ 16(m^2 + 3m + 21) \][/tex]
The polynomial inside the parentheses, [tex]\(m^2 + 3m + 21\)[/tex], does not factor further over the real numbers, so our simplification stops here.
Hence, the detailed, step-by-step solutions for each part:
1. Solutions for [tex]\(11x^2 = 8x + 9\)[/tex]:
[tex]\[ x = \frac{4 - \sqrt{115}}{11} \quad \text{and} \quad x = \frac{4 + \sqrt{115}}{11} \][/tex]
2. Solutions for [tex]\(2y^2 = 20y + 100\)[/tex]:
[tex]\[ y = 5 - 5\sqrt{3} \quad \text{and} \quad y = 5 + 5\sqrt{3} \][/tex]
3. Simplified expression for [tex]\(4x^2 + 16xy + 16y^2\)[/tex]:
[tex]\[ 4(x + 2y)^2 \][/tex]
4. Simplified expression for [tex]\(16a^2 - 240x + 4b^2\)[/tex]:
[tex]\[ 16a^2 + 4b^2 - 240x \][/tex]
5. Simplified expression for [tex]\(16m^2 + 48m + 336\)[/tex]:
[tex]\[ 16(m^2 + 3m + 21) \][/tex]
### 1. Solve the equation [tex]\(11x^2 = 8x + 9\)[/tex]
First, let's rearrange the equation to standard quadratic form:
[tex]\[ 11x^2 - 8x - 9 = 0 \][/tex]
We use the quadratic formula to find the roots of the equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\(a = 11\)[/tex], [tex]\(b = -8\)[/tex], and [tex]\(c = -9\)[/tex]. Plugging these values into the quadratic formula gives:
[tex]\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(11)(-9)}}{2(11)} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{64 + 396}}{22} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{460}}{22} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{115 \times 4}}{22} \][/tex]
[tex]\[ x = \frac{8 \pm 2\sqrt{115}}{22} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{115}}{11} \][/tex]
So, the solutions are:
[tex]\[ x = \frac{4 - \sqrt{115}}{11} \quad \text{and} \quad x = \frac{4 + \sqrt{115}}{11} \][/tex]
### 2. Solve the equation [tex]\(2y^2 = 20y + 100\)[/tex]
First, let's rearrange the equation to standard quadratic form:
[tex]\[ 2y^2 - 20y - 100 = 0 \][/tex]
Divide every term by 2:
[tex]\[ y^2 - 10y - 50 = 0 \][/tex]
Using the quadratic formula:
[tex]\[ y = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(-50)}}{2(1)} \][/tex]
[tex]\[ y = \frac{10 \pm \sqrt{100 + 200}}{2} \][/tex]
[tex]\[ y = \frac{10 \pm \sqrt{300}}{2} \][/tex]
[tex]\[ y = \frac{10 \pm 10\sqrt{3}}{2} \][/tex]
[tex]\[ y = 5 \pm 5\sqrt{3} \][/tex]
So, the solutions are:
[tex]\[ y = 5 - 5\sqrt{3} \quad \text{and} \quad y = 5 + 5\sqrt{3} \][/tex]
### 3. Simplify the expression [tex]\(4x^2 + 16xy + 16y^2\)[/tex]
The given expression can be rewritten as:
[tex]\[ 4(x^2 + 4xy + 4y^2) \][/tex]
Notice that inside the parenthesis is a perfect square trinomial:
[tex]\[ x^2 + 4xy + 4y^2 = (x + 2y)^2 \][/tex]
So, the expression simplifies to:
[tex]\[ 4(x + 2y)^2 \][/tex]
### 4. Simplify the expression [tex]\(16a^2 - 240x + 4b^2\)[/tex]
The expression [tex]\(16a^2 - 240x + 4b^2\)[/tex] is already in its simplified form. There is no further simplification since there are no common factors or like terms that can be combined.
### 5. Simplify the expression [tex]\(16m^2 + 48m + 336\)[/tex]
This expression is already in its expanded form. However, we can factor out the common factor of 16:
[tex]\[ 16(m^2 + 3m + 21) \][/tex]
The polynomial inside the parentheses, [tex]\(m^2 + 3m + 21\)[/tex], does not factor further over the real numbers, so our simplification stops here.
Hence, the detailed, step-by-step solutions for each part:
1. Solutions for [tex]\(11x^2 = 8x + 9\)[/tex]:
[tex]\[ x = \frac{4 - \sqrt{115}}{11} \quad \text{and} \quad x = \frac{4 + \sqrt{115}}{11} \][/tex]
2. Solutions for [tex]\(2y^2 = 20y + 100\)[/tex]:
[tex]\[ y = 5 - 5\sqrt{3} \quad \text{and} \quad y = 5 + 5\sqrt{3} \][/tex]
3. Simplified expression for [tex]\(4x^2 + 16xy + 16y^2\)[/tex]:
[tex]\[ 4(x + 2y)^2 \][/tex]
4. Simplified expression for [tex]\(16a^2 - 240x + 4b^2\)[/tex]:
[tex]\[ 16a^2 + 4b^2 - 240x \][/tex]
5. Simplified expression for [tex]\(16m^2 + 48m + 336\)[/tex]:
[tex]\[ 16(m^2 + 3m + 21) \][/tex]
