Solve The Following Quadratic Equation For All Values Of $x$ In Simplest Form:${(x+2)^2 + 25 = 50}$

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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, (x+2)2+25=50(x+2)^2 + 25 = 50, for all values of xx in simplest form.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable 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. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Given Quadratic Equation

The given quadratic equation is (x+2)2+25=50(x+2)^2 + 25 = 50. To solve this equation, we need to isolate the variable xx. The first step is to simplify the equation by subtracting 25 from both sides.

Simplifying the Equation

(x+2)2+25βˆ’25=50βˆ’25(x+2)^2 + 25 - 25 = 50 - 25

This simplifies to:

(x+2)2=25(x+2)^2 = 25

Solving for xx

To solve for xx, we need to take the square root of both sides of the equation. This will give us two possible solutions for xx.

Taking the Square Root

(x+2)2=25\sqrt{(x+2)^2} = \sqrt{25}

This simplifies to:

x+2=Β±5x+2 = \pm 5

Isolating xx

To isolate xx, we need to subtract 2 from both sides of the equation.

Subtracting 2

x+2βˆ’2=Β±5βˆ’2x+2 - 2 = \pm 5 - 2

This simplifies to:

x=βˆ’2Β±5x = -2 \pm 5

Simplifying the Solutions

We can simplify the solutions by combining the constants.

Simplifying the Solutions

x=βˆ’2+5x = -2 + 5 or x=βˆ’2βˆ’5x = -2 - 5

This simplifies to:

x=3x = 3 or x=βˆ’7x = -7

Conclusion

In this article, we solved the quadratic equation (x+2)2+25=50(x+2)^2 + 25 = 50 for all values of xx in simplest form. We simplified the equation, took the square root of both sides, and isolated xx to find the solutions. The final solutions are x=3x = 3 and x=βˆ’7x = -7.

Frequently Asked Questions

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.

Q: How do I solve a quadratic equation?

A: You can solve a quadratic equation using various methods, including factoring, completing the square, and 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=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Final Thoughts

Solving quadratic equations is an essential skill for students and professionals alike. By following the steps outlined in this article, you can solve quadratic equations with ease. Remember to simplify the equation, take the square root of both sides, and isolate xx to find the solutions. With practice and patience, you can become proficient in solving quadratic equations.

Additional Resources

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. 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 important topic.

Q&A: Quadratic Equations

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. 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: You can solve a quadratic equation using various methods, including factoring, completing the square, and the quadratic formula. The quadratic formula is a formula that can be used to solve quadratic equations. It is given by x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

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=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. This formula can be used to find the solutions to a quadratic equation when the equation cannot be factored.

Q: How do I factor a quadratic equation?

A: Factoring a quadratic equation involves expressing the equation as a product of two binomials. For example, the quadratic equation x2+5x+6x^2 + 5x + 6 can be factored as (x+3)(x+2)(x + 3)(x + 2).

Q: What is completing the square?

A: Completing the square is a method of solving quadratic equations by rewriting the equation in the form (x+a)2=b(x + a)^2 = b. This method involves adding and subtracting a constant term to create a perfect square trinomial.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of aa, bb, and cc into the formula. The formula is given by x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. You can then simplify the expression to find the solutions to the quadratic equation.

Q: What are the solutions to a quadratic equation?

A: The solutions to a quadratic equation are the values of xx that satisfy the equation. These solutions can be real or complex numbers.

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

A: The number of solutions to a quadratic equation can be determined by the discriminant, which is the expression b2βˆ’4acb^2 - 4ac. If the discriminant is positive, the equation has two 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: What is the discriminant?

A: The discriminant is the expression b2βˆ’4acb^2 - 4ac. It is used to determine the number of solutions to a quadratic equation.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use the x-intercepts and the vertex of the parabola. The x-intercepts are the points where the parabola intersects the x-axis, and the vertex is the point where the parabola is at its maximum or minimum value.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. By understanding the different methods of solving quadratic equations, including factoring, completing the square, and the quadratic formula, you can better solve these equations and apply them to real-world problems.

Additional Resources

Final Thoughts

Solving quadratic equations is an essential skill for students and professionals alike. By understanding the different methods of solving quadratic equations, including factoring, completing the square, and the quadratic formula, you can better solve these equations and apply them to real-world problems. Remember to practice and review the concepts to become proficient in solving quadratic equations.