Solve $3x^2 - 2x - 7 = X - 18$ For $x$ In The Form $ A + B I A + Bi A + Bi [/tex].

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a quadratic equation of the form $3x^2 - 2x - 7 = x - 18$ for $x$ in the form $a + bi$. We will break down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions for $x$ are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula is derived from the fact that the quadratic equation can be rewritten as a perfect square trinomial.

Solving the Given Equation

Now, let's apply the quadratic formula to solve the given equation $3x^2 - 2x - 7 = x - 18$. First, we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$.

3x2−2x−7−x+18=03x^2 - 2x - 7 - x + 18 = 0

Combine like terms:

3x2−3x+11=03x^2 - 3x + 11 = 0

Now, we can identify the values of $a$, $b$, and $c$:

a = 3$, $b = -3$, and $c = 11

Applying the Quadratic Formula

Substitute the values of $a$, $b$, and $c$ into the quadratic formula:

x=−(−3)±(−3)2−4(3)(11)2(3)x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(3)(11)}}{2(3)}

Simplify the expression:

x=3±9−1326x = \frac{3 \pm \sqrt{9 - 132}}{6}

x=3±−1236x = \frac{3 \pm \sqrt{-123}}{6}

Simplifying the Expression

The expression $\sqrt{-123}$ can be simplified using the fact that $\sqrt{-1} = i$, where $i$ is the imaginary unit.

−123=−1⋅123=−1⋅123=i123\sqrt{-123} = \sqrt{-1 \cdot 123} = \sqrt{-1} \cdot \sqrt{123} = i\sqrt{123}

Substitute this expression back into the solution:

x=3±i1236x = \frac{3 \pm i\sqrt{123}}{6}

Rationalizing the Denominator

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:

x=3±i1236⋅66x = \frac{3 \pm i\sqrt{123}}{6} \cdot \frac{6}{6}

x=3±i12336x = \frac{3 \pm i\sqrt{123}}{36}

Conclusion

In this article, we solved the quadratic equation $3x^2 - 2x - 7 = x - 18$ for $x$ in the form $a + bi$. We used the quadratic formula to arrive at the solution, and then simplified the expression using algebraic manipulations. The final answer is $x = \frac{3 \pm i\sqrt{123}}{36}$.

Final Answer

x=3±i12336x = \frac{3 \pm i\sqrt{123}}{36}

Note

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we solved a quadratic equation of the form $3x^2 - 2x - 7 = x - 18$ for $x$ in the form $a + bi$. In this article, we will answer some frequently asked questions about quadratic equations and provide additional insights into solving them.

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 (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, completing the square, and the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations and is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What is the quadratic formula?

A: The quadratic formula is a formula that gives the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, substitute these values into the quadratic formula and simplify the expression.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a highest power of two, while a linear equation has a highest power of one.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula gives two possible values for $x$, and these values are the only solutions to the equation.

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

A: To determine the number of solutions to a quadratic equation, you need to examine the discriminant, which is the expression $b^2 - 4ac$ under the square root in the quadratic formula. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the discriminant?

A: The discriminant is the expression $b^2 - 4ac$ under the square root in the quadratic formula. It determines the number of solutions to a quadratic equation.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. If the discriminant is negative, the equation has no real solutions, but it may have complex solutions.

Conclusion

In this article, we answered some frequently asked questions about quadratic equations and provided additional insights into solving them. We hope that this article has been helpful in clarifying any doubts you may have had about quadratic equations.

Final Answer

The quadratic formula is a powerful tool for solving quadratic equations and is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Note

The quadratic formula can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The formula gives two possible values for $x$, and these values are the only solutions to the equation.