Square of sum: Equations with variables on both sides

To solve the equation $\sqrt{x+1} \times \sqrt{x+2} = x+3$, we will follow these steps:

• Step 1: Identify the terms. Let $a = \sqrt{x+1}$ and $b = \sqrt{x+2}$, so $a \cdot b = x + 3$.
• Step 2: Square both sides of the equation to remove the square roots: $(ab)^2 = (x+3)^2$.
• Step 3: Express using the original variables: $(x+1)(x+2) = (x+3)^2$.
• Step 4: Expand both sides:
  Left side: $x^2 + 3x + 2$
  Right side: $x^2 + 6x + 9$.
• Step 5: Rearrange and simplify the equation by subtracting one side from the other:
  $x^2 + 3x + 2 = x^2 + 6x + 9$ becomes $0 = 3x + 7$.
• Step 6: Solve the linear equation: $3x = -7$, hence $x = -\frac{7}{3}$.
• Step 7: Verify that the solution $x = -\frac{7}{3}$ fits the initial domain requirements for the square roots.

Therefore, the solution to the problem is $x = -\frac{7}{3}$. This matches choice 3 in the provided answer choices.

To solve the equation $13x^2 + 4x = 8(x+3)^2$, we proceed with the following steps:

• **Step 1: Expand the right-hand side.**
  We start by expanding $8(x+3)^2$:
  $(x+3)^2 = x^2 + 6x + 9$
  Multiplying by 8 gives $8(x^2 + 6x + 9) = 8x^2 + 48x + 72$.
• **Step 2: Form the standard quadratic equation.**
  Now substitute back into the initial equation:
  $13x^2 + 4x = 8x^2 + 48x + 72$
  Rearrange all terms to one side:
  $13x^2 + 4x - 8x^2 - 48x - 72 = 0$
  Simplify: $5x^2 - 44x - 72 = 0$.
• **Step 3: Apply the quadratic formula.**
  The equation is in the standard form $ax^2 + bx + c = 0$ where $a = 5$, $b = -44$, and $c = -72$.
  Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we calculate:
  $x = \frac{-(-44) \pm \sqrt{(-44)^2 - 4 \times 5 \times (-72)}}{2 \times 5}$
  $x = \frac{44 \pm \sqrt{1936 + 1440}}{10}$
  $x = \frac{44 \pm \sqrt{3376}}{10}$
  $\sqrt{3376} \approx 58.1$, so:
  $x = \frac{44 \pm 58.1}{10}$.
• **Step 4: Calculate the roots.**
  For the positive solution:
  $x_1 = \frac{44 + 58.1}{10} = 10.21$.
  For the negative solution:
  $x_2 = \frac{44 - 58.1}{10} = -1.41$.

Therefore, the solutions to the equation are $x_1 = 10.21$ and $x_2 = -1.41$. The correct choice from the given options is choice 3.

To solve the equation $7x + 1 + (2x + 3)^2 = (4x + 2)^2$, we follow these steps:

• Step 1: Expand both sides using the square of a binomial formula.
• Step 2: Simplify the equation to form a standard quadratic equation.
• Step 3: Use the quadratic formula to find the roots of the equation.

Step 1: Expand the squares.

The left side: $(2x + 3)^2 = 4x^2 + 12x + 9$.

The right side: $(4x + 2)^2 = 16x^2 + 16x + 4$.

Step 2: Substitute back into the original equation and simplify:

$7x + 1 + 4x^2 + 12x + 9 = 16x^2 + 16x + 4$.

Combine like terms:

$4x^2 + 19x + 10 = 16x^2 + 16x + 4$.

Step 3: Move all terms to one side:

$4x^2 + 19x + 10 - 16x^2 - 16x - 4 = 0$.

Which simplifies to:

$-12x^2 + 3x + 6 = 0$.

Step 4: Divide by -3 to simplify:

$4x^2 - x - 2 = 0$.

Step 5: Use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 4$, $b = -1$, $c = -2$.

Calculate the discriminant:

$b^2 - 4ac = (-1)^2 - 4 \cdot 4 \cdot (-2) = 1 + 32 = 33$.

Calculate the roots:

$x = \frac{1 \pm \sqrt{33}}{8}$.

Therefore, the solution to the problem is $x = \frac{1 \pm \sqrt{33}}{8}$.

To solve this problem, we'll follow these steps:

• Step 1: Cross-multiply the given equation to eliminate fractions.
• Step 2: Expand the squared terms on either side of the equation.
• Step 3: Rearrange terms to form a quadratic equation.
• Step 4: Solve the quadratic equation to find possible values for $x$.

Now, let's work through each step:

Step 1: Begin with the given equation:
$\frac{(\frac{1}{x} + \frac{1}{2})^2}{(\frac{1}{x} + \frac{1}{3})^2} = \frac{81}{64}$. Cross-multiply to eliminate fractions:
$(\frac{1}{x} + \frac{1}{2})^2 \times 64 = (\frac{1}{x} + \frac{1}{3})^2 \times 81$.

Step 2: Expand each squared term:
For $(\frac{1}{x} + \frac{1}{2})^2$, use $(a + b)^2 = a^2 + 2ab + b^2$:
$(\frac{1}{x})^2 + 2(\frac{1}{x})(\frac{1}{2}) + (\frac{1}{2})^2 = \frac{1}{x^2} + \frac{1}{x} + \frac{1}{4}$.
Similarly, $(\frac{1}{x} + \frac{1}{3})^2 = \frac{1}{x^2} + \frac{2}{3x} + \frac{1}{9}$.

Step 3: Substitute these into the cross-multiplied equation:
$64\left(\frac{1}{x^2} + \frac{1}{x} + \frac{1}{4}\right) = 81\left(\frac{1}{x^2} + \frac{2}{3x} + \frac{1}{9}\right)$.

Step 4: Simplify and collect like terms:
$64(\frac{1}{x^2} + \frac{1}{x} + \frac{1}{4}) = (64\frac{1}{x^2} + 64\frac{1}{x} + 16)$,
$81(\frac{1}{x^2} + \frac{2}{3x} + \frac{1}{9}) = (81\frac{1}{x^2} + 54\frac{1}{x} + 9)$.

Equating terms gives:
$64\frac{1}{x^2} + 64\frac{1}{x} + 16 = 81\frac{1}{x^2} + 54\frac{1}{x} + 9$.

Step 5: Solve the quadratic equation:
Combine like terms: $-17\frac{1}{x^2} + 10\frac{1}{x} + 7 = 0$.
Let $y = \frac{1}{x}$. Substitute to get: $-17y^2 + 10y + 7 = 0$.
Multiply the entire equation by -1 to simplify: $17y^2 - 10y - 7 = 0$.

Using the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=17$, $b=-10$, $c=-7$:
$y = \frac{10 \pm \sqrt{(-10)^2 - 4 \times 17 \times (-7)}}{2 \times 17}$
$y = \frac{10 \pm \sqrt{100 + 476}}{34}$
$y = \frac{10 \pm \sqrt{576}}{34}$
$y = \frac{10 \pm 24}{34}$
Which gives:
$y = \frac{34}{34} = 1$ or $y = -\frac{14}{34} = -\frac{7}{17}$.

Since $y = \frac{1}{x}$:
For $y=1$, $x = \frac{1}{y} = 1$.
For $y=-\frac{7}{17}$, $x = \frac{1}{y} = -\frac{17}{7}$.

Therefore, the solutions for $x$ are $x = 1$ and $x = -\frac{17}{7}$.

Checking the correct answer choice, these correspond to the second choice.

Thus, the solution to the problem is $x = 1, -\frac{17}{7}$.

To solve the equation $\frac{x^3 + 1}{(x+1)^2} = x$, we will follow these steps:

• Step 1: Set up the equation for solving by cross-multiplying.
• Step 2: Simplify and solve the resulting polynomial equation.
• Step 3: Solve for the values of $x$.

Let's work through the solution:

Step 1: Cross-multiply to eliminate the fraction:

Expand the right-hand side:

Step 2: Set the expanded equation equal:

Cancel $x^3$ from both sides:

Re-arrange the equation to form a standard quadratic equation:

Step 3: Solve the quadratic equation using the quadratic formula:

The quadratic formula is:

Substitute the values of $a$, $b$, and $c$ into the formula:

Calculate the discriminant and simplify:

Simplify further:

This gives the solutions:

Since $x = -1$ would make the denominator zero, it is not allowed as a solution. Thus, the only valid solution is:

Therefore, the solution to the equation is $x = \frac{1}{2}$.

To determine the correct relationship between $x$ and $y$, let's transform each equation:

Step 1: Transform the First Equation

The first equation is $x^2 + 4 = -6y$. Rearranging gives us:

$x^2 = -6y - 4$

Now, aim to complete the square for expressions involving $x$ and $y$.

Step 2: Transform the Second Equation

The second equation is $y^2 + 9 = -4x$. Rearranging gives us:

$y^2 = -4x - 9$

Step 3: Complete the Square

Let's complete the square for the terms $x^2$ and $y^2$.

For $x^2 = -6y - 4$:

$x^2 + 4x + 4 = -6y$

Thus, it becomes:

$(x + 2)^2 = -6y$

For $y^2 = -4x - 9$:

$y^2 + 6y + 9 = -4x$

Thus, it becomes:

$(y + 3)^2 = -4x$

Step 4: Combine and Analyze

Substitute back to express a sum of squares:

Adding these completes the square:

$(x + 2)^2 + (y + 3)^2 = 0$

This result shows that both squares, squared terms are zero-sum, revealing the conditions under which equations balance.

Thus, the correct choice according to the transformations conducted is:

$(x+2)^2 + (y+3)^2 = 0$

Therefore, the solution to the problem is $(x+2)^2 + (y+3)^2 = 0$.

To solve the equation $(-x+1)^2=(2x+1)^2$, we will follow these steps:

• Step 1: Recognize the equation as an application of the identity $a^2 = b^2$. This implies $a = b$ or $a = -b$.
• Step 2: Apply the identity to our equation.
• Step 3: Solve each of the resulting equations individually to find the possible values of $x$.

Now, let's perform each step in detail:

Step 1: We have the equation $(-x+1)^2 = (2x+1)^2$. According to the identity $a^2 = b^2$, we can set up the following cases:
Case 1: $-x + 1 = 2x + 1$,
Case 2: $-x + 1 = -(2x + 1)$.

Step 2: Solve Case 1:
From $-x + 1 = 2x + 1$, subtract 1 from both sides: $-x = 2x$.
Adding $x$ to both sides gives $0 = 3x$.
Divide by 3: $x = 0$.

Step 3: Solve Case 2:
From $-x + 1 = -(2x + 1)$, distribute the negative sign on the right: $-x + 1 = -2x - 1$.
Add $2x$ to both sides: $x + 1 = -1$.
Subtract 1 from both sides: $x = -2$.

Therefore, the solutions to the equation are $x_1=0$ and $x_2=-2$.

The correct answer is:

$x_1=0,x_2=-2$

To solve the equation $(x+3)^2 = 2x + 5$, we proceed as follows:

• Step 1: Expand the left side. Using the identity $(a+b)^2 = a^2 + 2ab + b^2$, we find:
  $(x+3)^2 = x^2 + 6x + 9$.

• Step 2: Set the equation to zero by moving all terms to one side:
  $x^2 + 6x + 9 = 2x + 5$
  Subtract $2x + 5$ from both sides:
  $x^2 + 6x + 9 - 2x - 5 = 0$
  This simplifies to:
  $x^2 + 4x + 4 = 0$.

• Step 3: Solve the quadratic equation $x^2 + 4x + 4 = 0$. Notice this can be factored as:
  $(x+2)^2 = 0$.

• Step 4: Solve for $x$ by setting the factor equal to zero:
  $x+2 = 0$.
  Thus, $x = -2$.

Therefore, the solution to the equation $(x+3)^2 = 2x + 5$ is $x = -2$.

The given equation is:

$2x^2 - 2x = (x+1)^2$

Step 1: Expand the right-hand side.

$(x+1)^2 = x^2 + 2x + 1$

Step 2: Write the full equation with the expanded form.

$2x^2 - 2x = x^2 + 2x + 1$

Step 3: Bring all terms to one side of the equation to set it to zero.

$2x^2 - 2x - x^2 - 2x - 1 = 0$

Step 4: Simplify the equation.

$x^2 - 4x - 1 = 0$

Step 5: Identify coefficients for the quadratic formula.

Here, $a = 1$, $b = -4$, $c = -1$.

Step 6: Apply the quadratic formula.

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}$

$x = \frac{4 \pm \sqrt{16 + 4}}{2}$

$x = \frac{4 \pm \sqrt{20}}{2}$

$x = \frac{4 \pm 2\sqrt{5}}{2}$

$x = 2 \pm \sqrt{5}$

Therefore, the solutions are $x = 2 + \sqrt{5}$ and $x = 2 - \sqrt{5}$.

These solutions correspond to choice (4): Answers a + b

In order to solve the equation, start by removing the denominators.

To do this, we'll multiply the denominators:

$(2x+1)^2\cdot(2x+1)+(x+2)^2\cdot(x+2)=4.5x(2x+1)(x+2)$

Open the parentheses on the left side, making use of the distributive property:

$(4x^2+4x+1)\cdot(2x+1)+(x^2+4x+4)\cdot(x+2)=4.5x(2x+1)(x+2)$

Continue to open the parentheses on the right side of the equation:

$(4x^2+4x+1)\cdot(2x+1)+(x^2+4x+4)\cdot(x+2)=4.5x(2x^2+5x+2)$

Simplify further:

$(4x^2+4x+1)\cdot(2x+1)+(x^2+4x+4)\cdot(x+2)=9x^3+22.5x+9x$

Go back and simplify the parentheses on the left side of the equation:

$8x^3+8x^2+2x+4x^2+4x+1+x^3+4x^2+4x+2x^2+8x+8=9x^3+22.5x+9x$

Combine like terms:

$9x^3+18x^2+18x+9=9x^3+22.5x+9x$

Notice that all terms can be divided by 9 as shown below:

$x^3+2x^2+2x+1=x^3+2.5x+x$

Move all numbers to one side:

$x^3-x^3+2x^2-2.5x^2+2x-x+9=0$

We obtain the following:

$0.5x^2-x-1=0$

In order to remove the one-half coefficient, multiply the entire equation by 2

$x^2-2x-2=0$

Apply the square root formula, as shown below-

$\frac{2±\sqrt{12}}{2}$

Apply the properties of square roots in order to simplify the square root of 12:

$\frac{2±2\sqrt{3}}{2}$Divide both the numerator and denominator by 2 as follows:

$1±\sqrt{3}$

To solve the given equation $ax^2 + 5a + x = (3+a)x^2 - (x+a)^2$, we begin with expansion and simplification:

• Step 1: Expand the terms on the right side:
  $(3+a)x^2 - (x+a)^2 = (3+a)x^2 - (x^2 + 2ax + a^2) = (3+a)x^2 - x^2 - 2ax - a^2$
• Step 2: Simplify further:
  $(3+a)x^2 - x^2 - 2ax - a^2 = 2ax^2 - 2ax - a^2 + 3x^2$
• Step 3: Collect and equate coefficients from both sides:
  $0 = (2a + 2a - a)x^2 + (a - 5)x - a^2$
• Step 4: Set each type of coefficient separately to zero, assuming that the equation is valid for all $x$:
  - Coefficient of $x^2$: $a = 3$
  - Coefficient of $x$: $1 = 2a - 1$
  Solving these inequalities in terms of $a$ gives us the final inequality solution.

From the analysis, the solution is constrained by the inequalities derived from the simplification process. Hence, the answer is:
Thus, the solution to the problem is $-3.644 \ge a, -0.023 \le a$.