Find All Solutions To The Equation $a^2 - 26 = 10 - 3a^2$.

$$

Introduction to the Problem

The given equation is a quadratic equation in the variable $a$. The equation is $a^2 - 26 = 10 - 3a^2$. Our goal is to find all possible solutions for the variable $a$ that satisfy this equation. To do this, we will first simplify the equation and then use algebraic techniques to solve for $a$.

Simplifying the Equation

The first step in solving the equation is to simplify it by combining like terms. We can start by moving all the terms involving $a^2$ to one side of the equation and the constant terms to the other side.

$$a^2 - 26 = 10 - 3a^2$$

Subtracting $10$ from both sides gives:

$$a^2 - 36 = -3a^2$$

Adding $3a^2$ to both sides gives:

$$4a^2 - 36 = 0$$

Solving the Quadratic Equation

Now that we have simplified the equation, we can solve for $a$. The equation is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 4$, $b = 0$, and $c = -36$. We can use the quadratic formula to solve for $a$:

$$a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Since $b = 0$, the equation simplifies to:

$$a = \frac{\pm \sqrt{b^2 - 4ac}}{2a}$$

Substituting $a = 4$, $b = 0$, and $c = -36$ into the equation gives:

$$a = \frac{\pm \sqrt{0 - 4(4)(-36)}}{2(4)}$$

Simplifying the equation gives:

$$a = \frac{\pm \sqrt{576}}{8}$$

$$a = \frac{\pm 24}{8}$$

Finding the Solutions

Now that we have simplified the equation, we can find the solutions for $a$. We have two possible solutions:

$$a = \frac{24}{8}$$

$$a = -\frac{24}{8}$$

Simplifying the solutions gives:

$$a = 3$$

$$a = -3$$

Conclusion

In this article, we have found all the solutions to the equation $a^2 - 26 = 10 - 3a^2$. We first simplified the equation by combining like terms and then used the quadratic formula to solve for $a$. The solutions to the equation are $a = 3$ and $a = -3$.

Final Answer

The final answer is $\boxed{3}$ and $\boxed{-3}$.

Additional Information

• The equation $a^2 - 26 = 10 - 3a^2$ is a quadratic equation in the variable $a$.
• The solutions to the equation are $a = 3$ and $a = -3$.
• The quadratic formula is a useful tool for solving quadratic equations.
• The quadratic formula is given by $a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Related Topics

• Quadratic equations
• Quadratic formula
• Algebra
• Mathematics

References

• Quadratic Formula
• Quadratic Equations
• Algebra
• Mathematics
  

$$

Q&A: Solutions to the Equation $a^2 - 26 = 10 - 3a^2$

Q: What is the equation $a^2 - 26 = 10 - 3a^2$?

A: The equation $a^2 - 26 = 10 - 3a^2$ is a quadratic equation in the variable $a$. It is a mathematical equation that involves a squared variable and can be solved using algebraic techniques.

Q: How do I simplify the equation $a^2 - 26 = 10 - 3a^2$?

A: To simplify the equation, we need to combine like terms. We can start by moving all the terms involving $a^2$ to one side of the equation and the constant terms to the other side. This will give us a simpler equation that we can solve.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by $a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula can be used to find the solutions to a quadratic equation.

Q: How do I use the quadratic formula to solve the equation $a^2 - 26 = 10 - 3a^2$?

A: To use the quadratic formula, we need to identify the values of $a$, $b$, and $c$ in the equation. In this case, $a = 4$, $b = 0$, and $c = -36$. We can then substitute these values into the quadratic formula and simplify to find the solutions.

Q: What are the solutions to the equation $a^2 - 26 = 10 - 3a^2$?

A: The solutions to the equation are $a = 3$ and $a = -3$. These are the values of $a$ that satisfy the equation.

Q: Why is it important to simplify the equation before solving it?

A: Simplifying the equation before solving it can make it easier to find the solutions. By combining like terms and isolating the variable, we can make the equation more manageable and increase our chances of finding the correct solutions.

Q: Can I use other methods to solve the equation $a^2 - 26 = 10 - 3a^2$?

A: Yes, there are other methods that can be used to solve the equation. For example, we can use factoring or the square root method to find the solutions. However, the quadratic formula is a powerful tool that can be used to solve quadratic equations, and it is often the most efficient method.

Q: What is the final answer to the equation $a^2 - 26 = 10 - 3a^2$?

A: The final answer to the equation is $a = 3$ and $a = -3$. These are the values of $a$ that satisfy the equation.

Additional Q&A

• Q: What is the difference between a quadratic equation and a linear equation? A: A quadratic equation is a mathematical equation that involves a squared variable, while a linear equation is a mathematical equation that involves a single variable.
• Q: How do I know if an equation is quadratic or linear? A: To determine if an equation is quadratic or linear, we need to look at the highest power of the variable. If the highest power is 2, then the equation is quadratic. If the highest power is 1, then the equation is linear.
• Q: Can I use the quadratic formula to solve any quadratic equation? A: Yes, the quadratic formula can be used to solve any quadratic equation. However, we need to make sure that the equation is in the correct form and that we have the correct values for $a$, $b$, and $c$.
• Q: What is the significance of the quadratic formula in mathematics? A: The quadratic formula is a powerful tool that can be used to solve quadratic equations. It is a fundamental concept in mathematics and is used in a wide range of applications, including physics, engineering, and computer science.

Related Topics

• Quadratic equations
• Quadratic formula
• Algebra
• Mathematics

References

• Quadratic Formula
• Quadratic Equations
• Algebra
• Mathematics