Extracting Square Roots: Solving the equation

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

• Step 1: Recognize the structure of the equation as a difference of squares.
• Step 2: Factor the equation.
• Step 3: Use the zero-product property to find the values of $x$.

Now, let's work through each step:
Step 1: The equation given is $x^2 - 16 = 0$. This equation is a type of difference of squares, as it can be expressed in the form $a^2 - b^2 = (a - b)(a + b)$.
Step 2: Recognizing $16$ as $4^2$, we can factor the equation: $x^2 - 16 = (x - 4)(x + 4) = 0$.
Step 3: According to the zero-product property, if the product of two expressions is zero, then at least one of the expressions must be zero. Therefore, we set each factor equal to zero:
$x - 4 = 0$ or $x + 4 = 0$.
Solving these simple linear equations gives $x = 4$ and $x = -4$.

Therefore, the solutions to the equation are $x_1 = -4$ and $x_2 = 4$.

To solve the equation $x^2 - 25 = 0$, follow these steps:

• Step 1: Isolate the square term: Begin by rewriting the equation to isolate $x^2$:

$x^2 - 25 = 0$
Add $25$ to both sides to obtain:
$x^2 = 25$

• Step 2: Extract the square roots: To solve $x^2 = 25$, take the square root of both sides. Remember to consider both the positive and negative roots:

$x = \pm \sqrt{25}$

• Step 3: Simplify: Calculate the square root of 25:

$\sqrt{25} = 5$

Therefore, the solutions are:
$x = 5$
and
$x = -5$

Thus, the solutions to the equation $x^2 - 25 = 0$ are $x_1 = 5$ and $x_2 = -5$.

Verifying with the provided choices, the correct choice matches the solution $x_1 = 5, x_2 = -5$.

Therefore, the solution to the problem is $x_1 = 5, x_2 = -5$.

To solve the equation $3x^2 - 12 = 0$, follow these steps:

• Step 1: Add 12 to both sides to begin isolating $x$.
• Step 2: The equation is now $3x^2 = 12$.
• Step 3: Divide both sides by 3 to solve for $x^2$, giving us $x^2 = 4$.
• Step 4: Take the square root of both sides, keeping in mind the principle of the plus and minus roots: $x = \pm \sqrt{4}$.
• Step 5: Calculate the square root: $x = \pm 2$.

Thus, the solutions to the quadratic equation are $x_1 = 2$ and $x_2 = -2$.

Therefore, the solution to the problem is $x_1 = 2, x_2 = -2$.

First, we should notice that it is a quadratic equation because there is a quadratic term (meaning raised to the second power).

The first step in solving a quadratic equation is always arranging it in a form where all terms on one side are ordered from highest to lowest power (in descending order from left to right) and 0 on the other side.

Then we can choose whether to solve it using the quadratic formula or by factoring/completing the square.

The equation in the problem is already arranged, so let's proceed with the solving technique:

We'll choose to solve it using the quadratic formula.

Let's recall it first:

The rule states that the roots of an equation in the form$ax^2+bx+c=0$ are $x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.

$$This formula is called: "The Quadratic Formula"

Let's now solve the problem:

$4x^2+1=0$

First, let's identify the coefficients of the terms:

$\begin{cases}a=4 \\ b=0 \\ c=1\end{cases}$

Note that in the given equation there is no first-power term, so from comparing to the general form:

$ax^2+bx+c=0$

we can conclude that the coefficient $b$ (which is the coefficient of the first-power term $x$ in the general form) is 0.

Let's continue and get the equation's solutions (roots) by substituting the coefficients we noted earlier in the quadratic formula:

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{0\pm\sqrt{0^2-4\cdot4\cdot1}}{2\cdot4}$

Let's continue and calculate the expression under the root and simplify the expression:

$x_{1,2}=\frac{\pm\sqrt{-16}}{8}$

We now have a negative expression under the root and since we cannot extract a real root from a negative number, this equation has no real solutions.

In other words, there is no real value of $x$ that when substituted in the equation will give a true statement.

Therefore, the correct answer is answer D.

To solve this quadratic equation, we will use factoring.

Consider the given equation:
$-2x^2 + 4x = 0$

Step 1: Factor out the greatest common factor from the terms.

The common factor of $-2x^2$ and $4x$ is $2x$. Factoring this out gives:
$2x(-x + 2) = 0$

Step 2: Set each factor to zero.

• $2x = 0$
• $-x + 2 = 0$

Step 3: Solve each equation.

• For $2x = 0$:
  Divide both sides by 2, resulting in: $x = 0$.
• For $-x + 2 = 0$:
  Add $x$ to both sides, giving $2 = x$ or $x = 2$.

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

The correct answer is choice 4: $x_1 = 2, x_2 = 0$.

To solve the quadratic equation $2x^2 - 8 = 0$, we will follow these steps:

• Step 1: Isolate $x^2$.
• Step 2: Solve for $x$ by taking square roots.

Let's perform each step:

Step 1: Isolate $x^2$
Start with the given equation:
$2x^2 - 8 = 0$

Add 8 to both sides to isolate the term involving $x$:
$2x^2 = 8$

Divide both sides by 2 to solve for $x^2$:
$x^2 = 4$

Step 2: Solve for $x$ by taking square roots
Take the square root of both sides, remembering to consider both the positive and negative roots:
$x = \pm \sqrt{4}$

Simplify the square root:
$x = \pm 2$

This means there are two solutions:
$x_1 = 2$ and $x_2 = -2$

Therefore, the solutions to the equation $2x^2 - 8 = 0$ are $x_1 = -2, x_2 = 2$.

Matching this with the choices provided, the correct answer is choice 3: $x_1 = -2, x_2 = 2$.