Solve For \[$ X \$\].$\[ 7x^2 + 10 = 18 \\]

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Introduction

In this article, we will be solving a quadratic equation to find the value of $x$. The equation given is $7x^2 + 10 = 18$. Our goal is to isolate the variable $x$ and find its value. We will use algebraic methods to solve this equation.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In this case, the equation is $7x^2 + 10 = 18$. To make it a standard quadratic equation, we need to move the constant term to the right-hand side of the equation.

Rearranging the Equation

To rearrange the equation, we need to subtract $10$ from both sides of the equation. This will give us:

$$7x^2 = 18 - 10$$

$$7x^2 = 8$$

Isolating the Variable

Now that we have the equation in the form of $7x^2 = 8$, we need to isolate the variable $x$. To do this, we need to divide both sides of the equation by $7$. This will give us:

$$x^2 = \frac{8}{7}$$

Taking the Square Root

To find the value of $x$, we need to take the square root of both sides of the equation. This will give us:

$$x = \pm \sqrt{\frac{8}{7}}$$

Simplifying the Expression

The expression $\sqrt{\frac{8}{7}}$ can be simplified by taking the square root of the numerator and the denominator separately. This will give us:

$$x = \pm \frac{\sqrt{8}}{\sqrt{7}}$$

Rationalizing the Denominator

To rationalize the denominator, we need to multiply both the numerator and the denominator by $\sqrt{7}$. This will give us:

$$x = \pm \frac{\sqrt{8} \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}}$$

$$x = \pm \frac{\sqrt{56}}{7}$$

Simplifying the Expression Further

The expression $\sqrt{56}$ can be simplified by taking the square root of the numerator. This will give us:

$$x = \pm \frac{\sqrt{4 \cdot 14}}{7}$$

$$x = \pm \frac{2\sqrt{14}}{7}$$

Conclusion

In this article, we solved the quadratic equation $7x^2 + 10 = 18$ to find the value of $x$. We used algebraic methods to isolate the variable $x$ and found its value to be $\pm \frac{2\sqrt{14}}{7}$. This is the solution to the given equation.

Final Answer

The final answer is $\boxed{\pm \frac{2\sqrt{14}}{7}}$.

Related Topics

• Quadratic Equations
• Algebraic Methods
• Solving Equations

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Luca Trevisan

Further Reading

• [1] "Quadratic Equations: A Comprehensive Guide"
• [2] "Algebraic Methods for Solving Equations"
• [3] "Mathematics for Computer Science: A Textbook"
  

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Introduction

In our previous article, we solved the quadratic equation $7x^2 + 10 = 18$ to find the value of $x$. We used algebraic methods to isolate the variable $x$ and found its value to be $\pm \frac{2\sqrt{14}}{7}$. In this article, we will answer some frequently asked questions related to solving quadratic equations.

Q&A

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to isolate the variable $x$. You can do this by using algebraic methods such as factoring, completing the square, or using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and solve for $x$.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. The quadratic formula is a formula that can be used to solve any quadratic equation, while factoring is a method that involves finding the factors of the quadratic expression.

Q: Can I use the quadratic formula to solve all quadratic equations?

A: Yes, you can use the quadratic formula to solve all quadratic equations. However, it may not always be the easiest method to use.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula. It is given by $b^2 - 4ac$. The discriminant can be used to determine the nature of the solutions to the quadratic equation.

Q: What are the possible values of the discriminant?

A: The discriminant can be positive, negative, or zero. If the discriminant is positive, the quadratic equation has two distinct real solutions. If the discriminant is negative, the quadratic equation has no real solutions. If the discriminant is zero, the quadratic equation has one real solution.

Q: How do I determine the nature of the solutions to a quadratic equation?

A: To determine the nature of the solutions to a quadratic equation, you need to calculate the discriminant. If the discriminant is positive, the quadratic equation has two distinct real solutions. If the discriminant is negative, the quadratic equation has no real solutions. If the discriminant is zero, the quadratic equation has one real solution.

Conclusion

In this article, we answered some frequently asked questions related to solving quadratic equations. We discussed the quadratic formula, factoring, and the significance of the discriminant. We also provided examples of how to use the quadratic formula and how to determine the nature of the solutions to a quadratic equation.

Final Answer

The final answer is $\boxed{\pm \frac{2\sqrt{14}}{7}}$.

Related Topics

• Quadratic Equations
• Algebraic Methods
• Solving Equations

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Luca Trevisan

Further Reading

• [1] "Quadratic Equations: A Comprehensive Guide"
• [2] "Algebraic Methods for Solving Equations"
• [3] "Mathematics for Computer Science: A Textbook"