Solve For X X X . X 2 − 12 X + 36 = 0 X^2 - 12x + 36 = 0 X 2 − 12 X + 36 = 0 X = X = X =

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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 specific quadratic equation: x2−12x+36=0x^2 - 12x + 36 = 0. We will break down the solution step by step, using a combination of algebraic manipulations and mathematical concepts.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, xx) is two. The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

where aa, bb, and cc are constants. In our equation, a=1a = 1, b=−12b = -12, and c=36c = 36.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions 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 factored as:

(x−r)(x−s)=0(x - r)(x - s) = 0

where rr and ss are the roots of the equation. By expanding the product on the left-hand side, we get:

x2−(r+s)x+rs=0x^2 - (r + s)x + rs = 0

Comparing this with the original equation, we can see that r+s=−bar + s = -\frac{b}{a} and rs=cars = \frac{c}{a}.

Applying the Quadratic Formula

Now that we have the quadratic formula, we can apply it to our equation. Plugging in the values of aa, bb, and cc, we get:

x=−(−12)±(−12)2−4(1)(36)2(1)x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(36)}}{2(1)}

Simplifying the expression, we get:

x=12±144−1442x = \frac{12 \pm \sqrt{144 - 144}}{2}

x=12±02x = \frac{12 \pm \sqrt{0}}{2}

x=122x = \frac{12}{2}

x=6x = 6

Conclusion

In this article, we solved the quadratic equation x2−12x+36=0x^2 - 12x + 36 = 0 using the quadratic formula. We started by understanding the general form of a quadratic equation and the quadratic formula. We then applied the formula to our specific equation, simplifying the expression to find the solution. The final answer is x=6x = 6.

Tips and Variations

  • Factoring: In some cases, it may be possible to factor the quadratic equation directly. For example, the equation x2−12x+36=0x^2 - 12x + 36 = 0 can be factored as (x−6)2=0(x - 6)^2 = 0.
  • Graphing: Quadratic equations can also be solved graphically. By plotting the graph of the equation, we can find the x-intercepts, which correspond to the solutions.
  • Numerical Methods: In some cases, it may not be possible to find an exact solution using algebraic methods. In such cases, numerical methods such as the Newton-Raphson method can be used to approximate the solution.

Real-World Applications

Quadratic equations have numerous real-world applications in fields such as physics, engineering, and economics. For example:

  • Projectile Motion: The trajectory of a projectile under the influence of gravity can be modeled using quadratic equations.
  • Optimization: Quadratic equations can be used to optimize functions, such as finding the maximum or minimum value of a function.
  • Finance: Quadratic equations can be used to model financial instruments, such as options and futures.

Conclusion

In conclusion, solving quadratic equations is a fundamental skill that has numerous applications in mathematics and other fields. By understanding the quadratic formula and applying it to specific equations, we can find the solutions to a wide range of problems. Whether it's factoring, graphing, or using numerical methods, there are many ways to approach quadratic equations. With practice and patience, anyone can become proficient in solving these equations and unlock the secrets of mathematics.

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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 address some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this topic.

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, xx) is two. The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

where aa, bb, and cc are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including:

  • Factoring: If the equation can be factored, you can find the solutions by setting each factor equal to zero.
  • Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

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

  • Graphing: Quadratic equations can also be solved graphically. By plotting the graph of the equation, you can find the x-intercepts, which correspond to the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are 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 plug in the values of aa, bb, and cc into the formula. For example, if you have the equation x2−12x+36=0x^2 - 12x + 36 = 0, you can plug in a=1a = 1, b=−12b = -12, and c=36c = 36 into the formula.

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. Factoring involves finding the factors of the quadratic expression, while the quadratic formula involves using a formula to find the solutions.

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

A: Yes, the quadratic formula can be used to solve all quadratic equations. However, in some cases, it may be more difficult to apply the formula than to factor the equation.

Q: How do I know if a quadratic equation can be factored?

A: A quadratic equation can be factored if it can be written in the form (x−r)(x−s)=0(x - r)(x - s) = 0, where rr and ss are the roots of the equation.

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, which is b2−4acb^2 - 4ac. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: Can I use the quadratic formula to solve quadratic equations with complex solutions?

A: Yes, the quadratic formula can be used to solve quadratic equations with complex solutions. However, in this case, the solutions will be complex numbers.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the quadratic formula and applying it to specific equations, you can find the solutions to a wide range of problems. Whether it's factoring, graphing, or using numerical methods, there are many ways to approach quadratic equations. With practice and patience, anyone can become proficient in solving these equations and unlock the secrets of mathematics.