Solve Using the Square Root Property 6x^2+125=x^2

Solution:

Step 1: Isolate the variable terms on one side.

1. Subtract $x^2$ from both sides: $6x^2 + 125 - x^2 = 0$
2. Combine like terms: $5x^2 + 125 = 0$

Step 2: Move the constant to the other side.

• Subtract 125 from both sides: $5x^2 = -125$

Step 3: Normalize the coefficient of $x^2$.

1. Divide by 5: $\frac{5x^2}{5} = \frac{-125}{5}$
2. Simplify: $x^2 = -25$

Step 4: Apply the square root property.

• Take the square root of both sides: $x = \pm\sqrt{-25}$

Step 5: Simplify the square root of a negative number.

1. Express $-25$ as $-1(25)$: $x = \pm\sqrt{-1(25)}$
2. Separate the square roots: $x = \pm\sqrt{-1} \cdot \sqrt{25}$
3. Recognize the imaginary unit: $x = \pm i \cdot \sqrt{25}$
4. Simplify the square root of 25: $x = \pm i \cdot 5$
5. Rearrange the terms: $x = \pm 5i$

Step 6: Present the complete solution.

1. Positive solution: $x = 5i$
2. Negative solution: $x = -5i$
3. Combine both solutions: $x = 5i, -5i$

Knowledge Notes:

The problem involves solving a quadratic equation using the square root property. Here are the relevant knowledge points:

1. Quadratic Equations: An equation of the form $ax^2 + bx + c = 0$ is called a quadratic equation, where $a$, $b$, and $c$ are constants, and $a \neq 0$.

2. Square Root Property: If $x^2 = k$, then $x = \pm\sqrt{k}$. This property allows us to solve quadratic equations by taking the square root of both sides.

3. Imaginary Numbers: When dealing with the square root of a negative number, we introduce the imaginary unit $i$, where $i = \sqrt{-1}$. Thus, $\sqrt{-k} = \sqrt{k} \cdot i$ for any positive number $k$.

4. Simplifying Square Roots: The square root of a perfect square, such as $\sqrt{25}$, simplifies to the number whose square is the radicand, in this case, 5.

5. Combining Like Terms: When solving equations, we often combine terms with the same variable to simplify the equation.

6. Moving Terms Across the Equation: We can move terms from one side of the equation to the other by performing the opposite operation (addition to subtraction, multiplication to division) on both sides.

7. Complex Solutions: Quadratic equations can have complex solutions when the discriminant (the part under the square root in the quadratic formula) is negative. These solutions are of the form $a + bi$ or $a - bi$ where $a$ and $b$ are real numbers.

By following these steps and knowledge points, we can solve quadratic equations even when they result in complex solutions.