Mathos AI | Solve for X Calculator - Solve Any Equation for X

Introduction

Have you ever looked at an equation and wondered, "How do I solve for $x$ ?" You're not alone! Solving for $x$ is a fundamental skill in mathematics, especially in algebra, that opens the door to understanding more complex concepts. Whether you're balancing a budget, calculating the trajectory of a rocket, or simply trying to pass your next math test, knowing how to solve for $x$ is essential.

In this comprehensive guide, we'll break down the process of solving for $x$ in different types of equations:

• Linear Equations
• Quadratic Equations
• Polynomial Equations

We'll provide step-by-step explanations to make complex math concepts easy to understand, even for beginners. Plus, we'll introduce you to the Mathos AI Solve for $x$ Calculator, a powerful tool that simplifies calculations and helps you learn faster.

Keywords to keep in mind:

• $\quad$ Solve for $x$ calculator
• $\quad$ Solve for $x$

Let's dive in!

What Does It Mean to Solve for $\boldsymbol{x}$ ?

Understanding Variables and Equations

An equation is a mathematical statement that asserts the equality of two expressions. It consists of:

• Variables: Symbols like $x$ that represent unknown values.
• Constants: Known values like numbers.
• Operators: Mathematical operations like addition ($+$), subtraction ($-$), multiplication ( $\times$ ), and division ( $\div$ ).

Solving for $x$ means finding the value(s) of $x$ that make the equation true.

Why is this important?

• Foundation of Algebra: Solving for variables is a core skill in algebra.
• Real-World Applications: Used in fields like physics, engineering, economics, and more.
• Problem-Solving Skills: Enhances logical thinking and analytical abilities.

How to Solve for $x$ in Linear Equations

Understanding Linear Equations

A linear equation is an equation between two variables that gives a straight line when plotted graph. It has the general form: $$ a x+b=c $$

• $a, b$, and $c$ are constants.
• $x$ is the variable we need to solve for.

Characteristics of Linear Equations:

• The variable $x$ is raised to the power of 1.
• Graphically, it represents a straight line.
• There is only one solution for $x$.

Step-by-Step Guide to Solving Linear Equations

Example 1:

Solve for $x$ : $$ 3 x+5=14 $$

Step 1: Isolate the Variable Term

We want to get $x$ by itself on one side of the equation.

• Subtract 5 from both sides to eliminate the constant term on the left side. $$ 3 x+5-5=14-5 $$

Simplify: $$ 3 x=9 $$

Explanation: We perform the same operation on both sides to maintain the balance of the equation.

Step 2: Solve for $x$

• Divide both sides by 3 to isolate $x$. $$ \frac{3 x}{3}=\frac{9}{3} $$

Simplify: $$ x=3 $$

Answer: $x=3$

Explanation: By dividing, we isolate $x$ and find its value.

More Examples with Detailed Explanations

Example 2:

Solve for $x$ : $$ -2 x+7=1 $$

Step 1: Isolate the Variable Term

• Subtract 7 from both sides: $$ -2 x+7-7=1-7 $$

Simplify: $$ -2 x=-6 $$

Step 2: Solve for $x$

• Divide both sides by -2 : $$ \frac{-2 x}{-2}=\frac{-6}{-2} $$

Simplify: $$ x=3 $$

Answer: $x=3$

Example 3:

Solve for $x$ : $$ 5 x-4=2 x+8 $$

Step 1: Get All $x$ Terms on One Side

• Subtract $2 x$ from both sides: $$ 5 x-2 x-4=2 x-2 x+8 $$

Simplify: $$ 3 x-4=8 $$

Step 2: Isolate the Variable Term

• Add 4 to both sides: $$ 3 x-4+4=8+4 $$

Simplify: $$ 3 x=12 $$

Step 3: Solve for $x$

• Divide both sides by 3 : $$ \frac{3 x}{3}=\frac{12}{3} $$

Simplify: $$ x=4 $$

Answer: $x=4$

Using the Mathos Al Solve for $x$ Calculator for Linear Equations

The Mathos AI Solve for $x$ Calculator is a user-friendly tool that helps you solve linear equations quickly and understand each step.

How to Use It:

1. Enter the Equation:

• Type the equation into the calculator, e.g., $3 x+5=14$.

2. Click Calculate:

• The calculator processes the equation.

3. View the Solution:

• It displays the value of $x$ with step-by-step explanations.

Benefits:

• Instant Results: Get answers quickly.

• Step-by-Step Guidance: Understand how the solution is reached.

• Interactive Learning: Great for checking your work and learning the process.

How to Solve for $x$ in Quadratic Equations

Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation in one variable $x$, with the highest exponent being 2 .

Standard Form:

$$ a x^2+b x+c=0 $$

• $a, b$, and $c$ are constants (with $a \neq 0$ ).
• $x$ is the variable we need to solve for.

Characteristics:

• The graph of a quadratic equation is a parabola.
• There can be two, one, or no real solutions.

Methods to Solve Quadratic Equations

1. Factoring
2. Completing the Square
3. Quadratic Formula

We'll explore each method with examples.

Method 1: Solving by Factoring

When to Use: The quadratic equation can be factored into two binomials.

Example:

Solve for $x$ : $$ x^2-5 x+6=0 $$

Step 1: Factor the Quadratic

We need two numbers that multiply to +6 and add up to -5 .

• Possible pairs: $(-2,-3)$

Check: $$ \begin{gathered} (-2) \times(-3)=6 \ (-2)+(-3)=-5 \end{gathered} $$

Perfect!

Write the Factored Form: $$ (x-2)(x-3)=0 $$

Explanation: We express the quadratic as a product of two binomials.

Step 2: Set Each Factor to Zero

$$ x-2=0 \quad \text { or } \quad x-3=0 $$

Solve for $x$ :

• For $x-2=0$ : $$ x=2 $$
• For $x-3=0$ : $$ x=3 $$

Answer: $x=2$ or $x=3$

Explanation: Setting each factor to zero finds the values of $x$ that make the equation true.

Method 2: Solving by Completing the Square

When to Use: Useful when the quadratic cannot be easily factored.

Example:

Solve for $x$ : $$ x^2+6 x+5=0 $$

Step 1: Move the Constant Term to the Other Side

$$ x^2+6 x=-5 $$

Step 2: Find the Value to Complete the Square

• Take half of the coefficient of $x$, which is 6 : $$ \frac{6}{2}=3 $$

• Square it: $$ 3^2=9 $$

Step 3: Add the Square to Both Sides

$$ x^2+6 x+9=-5+9 $$

Simplify: $$ x^2+6 x+9=4 $$

Step 4: Write the Left Side as a Perfect Square

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

Explanation: The left side is now a squared binomial.

Step 5: Take the Square Root of Both Sides

$$ \sqrt{(x+3)^2}=\sqrt{4} $$

Simplify: $$ x+3= \pm 2 $$

Explanation: Remember to consider both positive and negative roots.

Step 6: Solve for $x$

• For $x+3=2$ : $$ x=2-3=-1 $$
• For $x+3=-2$ : $$ x=-2-3=-5 $$

Answer: $x=-1$ or $x=-5$

Method 3: Solving Using the Quadratic Formula

When to Use: Applicable to all quadratic equations.

Quadratic Formula:

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

Explanation of the Formula Components:

• $a$ : Coefficient of $x^2$

• $b$ : Coefficient of $x$

• $\quad c$ : Constant term

• $\sqrt{b^2-4 a c}$ : Discriminant; determines the nature of the roots.

Example:

Solve for $x$ : $$ 2 x^2-4 x-3=0 $$

Step 1: Identify $a, b$, and $c$

• $a=2$
• $b=-4$
• $c=-3$

Step 2: Plug Values into the Quadratic Formula

$$ x=\frac{-(-4) \pm \sqrt{(-4)^2-4 \times 2 \times(-3)}}{2 \times 2} $$

Step 3: Simplify the Expression

• Simplify numerator: $$ -(-4)=4 $$
• Calculate the discriminant: $$ (-4)^2-4 \times 2 \times(-3)=16+24=40 $$

Step 4: Write the Expression with the Simplified Discriminant

$$ x=\frac{4 \pm \sqrt{40}}{4} $$

Step 5: Simplify the Square Root

• $\sqrt{40}=\sqrt{4 \times 10}=2 \sqrt{10}$

Step 6: Simplify the Entire Expression

$$ x=\frac{4 \pm 2 \sqrt{10}}{4} $$

Simplify fractions: $$ \begin{aligned} x & =\frac{4}{4} \pm \frac{2 \sqrt{10}}{4} \ x & =1 \pm \frac{\sqrt{10}}{2} \end{aligned} $$

Answer: $$ x=1+\frac{\sqrt{10}}{2} \quad \text { or } \quad x=1-\frac{\sqrt{10}}{2} $$

Explanation: We have two real solutions involving square roots.

Using the Mathos AI Solve for $x$ Calculator for Quadratic Equations

The Mathos AI Solve for $x$ Calculator simplifies solving quadratic equations by handling all the calculations for you.

Benefits:

• Saves Time: No need to perform complex calculations manually.
• Accurate Results: Eliminates calculation errors.
• Educational: Helps you understand each step of the solution.

How to Solve for $x$ in Polynomial Equations

Understanding Polynomial Equations

A polynomial equation involves a polynomial expression set to zero. It can have degrees higher than two.

General Form:

$$ a_n x^n+a_{n-1} x^{n-1}+\ldots+a_1 x+a_0=0 $$

• $n$ is the highest power (degree) of $x$.
• $a_n, a_{n-1}, \ldots, a_0$ are constants.

Characteristics:

• May have multiple real or complex solutions.
• Degree $n$ indicates the maximum number of solutions.

Methods to Solve Polynomial Equations

1. Factoring
2. Rational Root Theorem
3. Synthetic Division
4. Graphical Methods

Method 1: Solving by Factoring

Example:

Solve for $x$ : $$ x^3-6 x^2+11 x-6=0 $$

Step 1: Factor the Polynomial

We look for factors that multiply to give the original polynomial.

Try to factor by grouping:

Group terms: $$ \left(x^3-6 x^2\right)+(11 x-6) $$

Factor out common terms: $$ x^2(x-6)+1(11 x-6) $$

This doesn't help directly, so let's look for rational roots using the Rational Root Theorem.

Step 2: Use Rational Root Theorem

Possible rational roots are factors of the constant term divided by factors of the leading coefficient.

• Factors of constant term (-6): $\pm 1, \pm 2, \pm 3, \pm 6$
• Leading coefficient is 1 , so factors are $\pm 1$

Possible roots: $\pm 1, \pm 2, \pm 3, \pm 6$

Step 3: Test Possible Roots

Test $x=1$ : $$ (1)^3-6(1)^2+11(1)-6=1-6+11-6=0 $$

Found a Root: $x=1$

Step 4: Factor Out $(x-1)$

Use polynomial division or synthetic division to divide the polynomial by $(x-1)$.

Resulting Polynomial: $$ (x-1)\left(x^2-5 x+6\right)=0 $$

Step 5: Factor the Quadratic

$$ x^2-5 x+6=(x-2)(x-3) $$

Step 6: Write the Fully Factored Form

$$ (x-1)(x-2)(x-3)=0 $$

Step 7: Solve for $x$

Set each factor to zero:

• $x-1=0 \Rightarrow x=1$
• $x-2=0 \Rightarrow x=2$
• $x-3=0 \Rightarrow x=3$

Answer: $x=1, x=2, x=3$

Method 2: Using the Rational Root Theorem and Synthetic Division

Example:

Solve for $x$ : $$ 2 x^3-3 x^2-8 x+12=0 $$

Step 1: Identify Possible Rational Roots

Factors of constant term (12): $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$ Factors of leading coefficient (2): $\pm 1, \pm 2$ Possible rational roots: $$ \pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{2}{1}, \pm \frac{2}{2}, \ldots $$

Simplify: $$ \pm 1, \pm \frac{1}{2}, \pm 2, \pm 3, \pm 6 $$

Step 2: Test Possible Roots Using Synthetic Division

Testing $x=2$ :

Remainder is $0$ , so $x=2$ is a root.

Step 3: Write the Depressed Polynomial

From synthetic division, the depressed polynomial is: $$ 2 x^2+x-6=0 $$

Step 4: Solve the Quadratic Equation

Use the quadratic formula: $$ x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} $$

With $a=2, b=1, c=-6$ : $$ x=\frac{-1 \pm \sqrt{1^2-4 \times 2 \times(-6)}}{2 \times 2} $$

Simplify: $$ \begin{gathered} x=\frac{-1 \pm \sqrt{1+48}}{4} \ x=\frac{-1 \pm \sqrt{49}}{4} \ x=\frac{-1^4 \pm 7}{4} \end{gathered} $$

Find the Solutions:

• $x=\frac{-1+7}{4}=\frac{6}{4}=\frac{3}{2}$

• $x=\frac{-1-7}{4}=\frac{-8}{4}=-2$

Step 5: List All Solutions

Including the root $x=2$ : Answer: $x=2, x=\frac{3}{2}, x=-2$

Using the Mathos AI Solve for $x$ Calculator for Polynomial Equations

The Mathos AI Solve for $x$ Calculator can handle higher-degree polynomial equations.

Benefits:

• Efficient: Quickly solves complex equations.
• Comprehensive: Handles multiple methods internally.
• Educational: Helps you understand the solving process.

Conclusion

Solving for $x$ is a fundamental skill in mathematics that applies to various types of equations, from simple linear ones to complex polynomials. By understanding the methods and practicing with different problems, you can master this skill and apply it in academic and real-world situations.

Key Takeaways:

• Linear Equations: Isolate $x$ by performing inverse operations.
• Quadratic Equations: Use factoring, completing the square, or the quadratic formula.
• Polynomial Equations: Factor when possible, use the Rational Root Theorem, and apply synthetic division.
• Practice: Regular practice enhances understanding and proficiency.
• Use Tools: The Mathos AI Solve for $x$ Calculator is an excellent resource for learning and verifying solutions.

Frequently Asked Questions

1. What does it mean to solve for $x$ ?

Solving for $x$ means finding the value(s) of $x$ that make the equation true. It's about determining the unknown variable in an equation.

2. How do I solve a linear equation for $x$ ?

• Step 1: Isolate the term containing $x$ by adding or subtracting terms on both sides.
• Step 2: Solve for $x$ by dividing or multiplying both sides by the coefficient of $x$.

3. When should I use the quadratic formula?

Use the quadratic formula when:

• The quadratic equation cannot be easily factored.
• You need exact solutions, especially when dealing with irrational numbers.

4. What is the discriminant in the quadratic formula?

The discriminant is $b^2-4 a c$ :

• If positive: Two real solutions.
• If zero: One real solution.
• If negative: No real solutions (but two complex solutions).

5. How does the Rational Root Theorem help in solving polynomial equations?

It provides a list of possible rational roots based on the factors of the constant term and the leading coefficient. Testing these roots helps identify actual solutions.

6. Can the Mathos AI Solve for $x$ Calculator handle complex equations?

Yes, the calculator is designed to handle linear, quadratic, and polynomial equations, providing step-by-step solutions.

7. Why is it important to learn different methods of solving equations?

Different equations may require different methods. Knowing multiple techniques allows you to choose the most efficient approach for a given problem.