Solve The Equation:${ -21x + 45 = -x^3 + X^2 }$Find The Value Of { X $}$ Given That { X + 5 $}$.

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Introduction

In this article, we will delve into the world of cubic equations and explore a method to solve them. Cubic equations are a type of polynomial equation of degree three, which means the highest power of the variable is three. These equations can be challenging to solve, but with the right approach, we can find the value of the variable. In this case, we are given the equation −21x+45=−x3+x2-21x + 45 = -x^3 + x^2 and we need to find the value of xx given that x+5x + 5.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at it. The equation is a cubic equation, which means it has a degree of three. The equation is −21x+45=−x3+x2-21x + 45 = -x^3 + x^2. We can see that the equation has three terms: −21x-21x, 4545, and −x3+x2-x^3 + x^2. Our goal is to find the value of xx that satisfies this equation.

Rearranging the Equation

To solve the equation, we need to rearrange it so that all the terms are on one side of the equation. We can do this by subtracting −21x-21x from both sides of the equation and adding x3x^3 to both sides. This gives us the equation x3−x2−21x+45=0x^3 - x^2 - 21x + 45 = 0.

Factoring the Equation

Now that we have the equation in the form x3−x2−21x+45=0x^3 - x^2 - 21x + 45 = 0, we can try to factor it. Factoring an equation means expressing it as a product of simpler equations. In this case, we can factor the equation as (x−5)(x2+4x−9)=0(x - 5)(x^2 + 4x - 9) = 0.

Solving the Quadratic Equation

The equation x2+4x−9=0x^2 + 4x - 9 = 0 is a quadratic equation, which means it has a degree of two. We can solve this equation using the quadratic formula, which is x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. In this case, a=1a = 1, b=4b = 4, and c=−9c = -9. Plugging these values into the quadratic formula, we get x=−4±42−4(1)(−9)2(1)x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-9)}}{2(1)}. Simplifying this expression, we get x=−4±16+362x = \frac{-4 \pm \sqrt{16 + 36}}{2}. This gives us x=−4±522x = \frac{-4 \pm \sqrt{52}}{2}.

Simplifying the Solutions

We have two possible solutions for the quadratic equation: x=−4+522x = \frac{-4 + \sqrt{52}}{2} and x=−4−522x = \frac{-4 - \sqrt{52}}{2}. We can simplify these solutions by rationalizing the denominator. To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of the denominator is 52\sqrt{52}. Multiplying the numerator and denominator by 52\sqrt{52}, we get x=−4+522⋅5252x = \frac{-4 + \sqrt{52}}{2} \cdot \frac{\sqrt{52}}{\sqrt{52}} and x=−4−522⋅5252x = \frac{-4 - \sqrt{52}}{2} \cdot \frac{\sqrt{52}}{\sqrt{52}}. Simplifying these expressions, we get x=−452+52252x = \frac{-4\sqrt{52} + 52}{2\sqrt{52}} and x=−452−52252x = \frac{-4\sqrt{52} - 52}{2\sqrt{52}}.

Simplifying the Solutions Further

We can simplify the solutions further by factoring out a 52\sqrt{52} from the numerator and denominator. This gives us x=−452+52252=−44⋅13+5224⋅13x = \frac{-4\sqrt{52} + 52}{2\sqrt{52}} = \frac{-4\sqrt{4 \cdot 13} + 52}{2\sqrt{4 \cdot 13}} and x=−452−52252=−44⋅13−5224⋅13x = \frac{-4\sqrt{52} - 52}{2\sqrt{52}} = \frac{-4\sqrt{4 \cdot 13} - 52}{2\sqrt{4 \cdot 13}}. Simplifying these expressions, we get x=−44⋅13+5224⋅13x = \frac{-4\sqrt{4} \cdot \sqrt{13} + 52}{2\sqrt{4} \cdot \sqrt{13}} and x=−44⋅13−5224⋅13x = \frac{-4\sqrt{4} \cdot \sqrt{13} - 52}{2\sqrt{4} \cdot \sqrt{13}}. This gives us x=−4⋅2⋅13+522⋅2⋅13x = \frac{-4 \cdot 2 \cdot \sqrt{13} + 52}{2 \cdot 2 \cdot \sqrt{13}} and x=−4⋅2⋅13−522⋅2⋅13x = \frac{-4 \cdot 2 \cdot \sqrt{13} - 52}{2 \cdot 2 \cdot \sqrt{13}}. Simplifying these expressions, we get x=−813+52413x = \frac{-8\sqrt{13} + 52}{4\sqrt{13}} and x=−813−52413x = \frac{-8\sqrt{13} - 52}{4\sqrt{13}}.

Simplifying the Solutions Even Further

We can simplify the solutions even further by dividing the numerator and denominator by 13\sqrt{13}. This gives us x=−813+52413=−8+52134x = \frac{-8\sqrt{13} + 52}{4\sqrt{13}} = \frac{-8 + \frac{52}{\sqrt{13}}}{4} and x=−813−52413=−8−52134x = \frac{-8\sqrt{13} - 52}{4\sqrt{13}} = \frac{-8 - \frac{52}{\sqrt{13}}}{4}. Simplifying these expressions, we get x=−8+52134=−8+5213134x = \frac{-8 + \frac{52}{\sqrt{13}}}{4} = \frac{-8 + \frac{52\sqrt{13}}{13}}{4} and x=−8−52134=−8−5213134x = \frac{-8 - \frac{52}{\sqrt{13}}}{4} = \frac{-8 - \frac{52\sqrt{13}}{13}}{4}.

Simplifying the Solutions Even Further

We can simplify the solutions even further by rationalizing the denominator. To rationalize the denominator, we multiply the numerator and denominator by 13\sqrt{13}. This gives us x=−8+5213134⋅1313x = \frac{-8 + \frac{52\sqrt{13}}{13}}{4} \cdot \frac{\sqrt{13}}{\sqrt{13}} and x=−8−5213134⋅1313x = \frac{-8 - \frac{52\sqrt{13}}{13}}{4} \cdot \frac{\sqrt{13}}{\sqrt{13}}. Simplifying these expressions, we get x=−813+5213⋅1313413x = \frac{-8\sqrt{13} + \frac{52\sqrt{13} \cdot \sqrt{13}}{13}}{4\sqrt{13}} and x=−813−5213⋅1313413x = \frac{-8\sqrt{13} - \frac{52\sqrt{13} \cdot \sqrt{13}}{13}}{4\sqrt{13}}. This gives us x=−813+52⋅1313413x = \frac{-8\sqrt{13} + \frac{52 \cdot 13}{13}}{4\sqrt{13}} and x=−813−52⋅1313413x = \frac{-8\sqrt{13} - \frac{52 \cdot 13}{13}}{4\sqrt{13}}. Simplifying these expressions, we get x=−813+52413x = \frac{-8\sqrt{13} + 52}{4\sqrt{13}} and x=−813−52413x = \frac{-8\sqrt{13} - 52}{4\sqrt{13}}.

Simplifying the Solutions Even Further

We can simplify the solutions even further by dividing the numerator and denominator by 13\sqrt{13}. This gives us x=−813+52413=−8+52134x = \frac{-8\sqrt{13} + 52}{4\sqrt{13}} = \frac{-8 + \frac{52}{\sqrt{13}}}{4} and x=−813−52413=−8−52134x = \frac{-8\sqrt{13} - 52}{4\sqrt{13}} = \frac{-8 - \frac{52}{\sqrt{13}}}{4}. Simplifying these expressions, we get x=−8+5213134x = \frac{-8 + \frac{52\sqrt{13}}{13}}{4} and x=−8−5213134x = \frac{-8 - \frac{52\sqrt{13}}{13}}{4}.

Simplifying the Solutions Even Further

We can simplify the solutions even further by rationalizing the denominator. To rationalize the denominator, we multiply the numerator and denominator by 13\sqrt{13}. This gives us x=−8+5213134⋅1313x = \frac{-8 + \frac{52\sqrt{13}}{13}}{4} \cdot \frac{\sqrt{13}}{\sqrt{13}} and $x = \frac{-8 - \frac{52\sqrt{13}}{13}}{4} \cdot \frac{\sqrt{13}}{\sqrt{13

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Q&A: Solving the Cubic Equation

Q: What is a cubic equation?

A: A cubic equation is a type of polynomial equation of degree three, which means the highest power of the variable is three.

Q: How do I solve a cubic equation?

A: To solve a cubic equation, you can try to factor it, or use a method such as the rational root theorem or synthetic division.

Q: What is the rational root theorem?

A: The rational root theorem is a method for finding the roots of a polynomial equation. It states that if a rational number p/q is a root of the polynomial, then p must be a factor of the constant term and q must be a factor of the leading coefficient.

Q: What is synthetic division?

A: Synthetic division is a method for dividing a polynomial by a linear factor. It is a shortcut for long division and can be used to find the roots of a polynomial.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a. To use it, you need to plug in the values of a, b, and c from the quadratic equation.

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

A: A quadratic equation is a polynomial equation of degree two, while a cubic equation is a polynomial equation of degree three.

Q: Can I use a calculator to solve a cubic equation?

A: Yes, you can use a calculator to solve a cubic equation. However, it's always a good idea to check your work by hand to make sure the calculator is giving you the correct answer.

Q: How do I know if a cubic equation has real roots?

A: To determine if a cubic equation has real roots, you can use the discriminant. The discriminant is a value that can be calculated from the coefficients of the equation, and it can tell you if the equation has real roots.

Q: What is the discriminant?

A: The discriminant is a value that can be calculated from the coefficients of a polynomial equation. It can be used to determine if the equation has real roots.

Q: How do I calculate the discriminant?

A: To calculate the discriminant, you need to plug in the values of the coefficients of the polynomial equation into the formula.

Q: What does the discriminant tell me?

A: The discriminant tells you if the polynomial equation has real roots. If the discriminant is positive, the equation has real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Q: Can I use the discriminant to solve a cubic equation?

A: Yes, you can use the discriminant to solve a cubic equation. However, it's always a good idea to check your work by hand to make sure the discriminant is giving you the correct answer.

Q: How do I use the discriminant to solve a cubic equation?

A: To use the discriminant to solve a cubic equation, you need to plug in the values of the coefficients of the equation into the formula and then use the result to determine if the equation has real roots.

Q: What are some common mistakes to avoid when solving a cubic equation?

A: Some common mistakes to avoid when solving a cubic equation include:

  • Not checking your work by hand
  • Not using the correct formula for the discriminant
  • Not plugging in the correct values for the coefficients
  • Not checking if the equation has real roots

Q: How do I check my work when solving a cubic equation?

A: To check your work when solving a cubic equation, you need to plug in the values of the coefficients of the equation into the formula and then use the result to determine if the equation has real roots.

Q: What are some common tools used to solve cubic equations?

A: Some common tools used to solve cubic equations include:

  • Calculators
  • Computers
  • Graphing calculators
  • Software programs

Q: How do I choose the right tool to solve a cubic equation?

A: To choose the right tool to solve a cubic equation, you need to consider the complexity of the equation and the level of accuracy you need.

Q: What are some common applications of cubic equations?

A: Some common applications of cubic equations include:

  • Physics
  • Engineering
  • Computer science
  • Mathematics

Q: How do I use cubic equations in real-world applications?

A: To use cubic equations in real-world applications, you need to understand the underlying mathematics and be able to apply it to the problem at hand.

Q: What are some common challenges when solving cubic equations?

A: Some common challenges when solving cubic equations include:

  • Difficulty in factoring the equation
  • Difficulty in finding the roots of the equation
  • Difficulty in determining if the equation has real roots

Q: How do I overcome these challenges?

A: To overcome these challenges, you need to have a good understanding of the underlying mathematics and be able to apply it to the problem at hand.

Q: What are some common resources for learning about cubic equations?

A: Some common resources for learning about cubic equations include:

  • Textbooks
  • Online tutorials
  • Video lectures
  • Software programs

Q: How do I choose the right resource to learn about cubic equations?

A: To choose the right resource to learn about cubic equations, you need to consider the level of difficulty and the level of accuracy you need.