Solve The Equation For X X X : X 2 + 70 X + 25 = 0 X^2 + 70x + 25 = 0 X 2 + 70 X + 25 = 0

Mar 12, 2025 by ADMIN 90 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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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, $x^2 + 70x + 25 = 0$ , using various methods. We will explore the different techniques and provide a step-by-step guide to help you understand the process.

What is a Quadratic Equation?

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, and $a$ cannot be zero.

The Quadratic Formula

One of the most common methods for solving quadratic equations is the quadratic formula. The quadratic formula is given by:

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

This formula can be used to solve any quadratic equation, regardless of the values of $a$ , $b$ , and $c$ .

Solving the Given Equation

Now, let's apply the quadratic formula to the given equation, $x^2 + 70x + 25 = 0$ . We have:

a = 1

b = 70

c = 25

Substituting these values into the quadratic formula, we get:

x = \frac{-70 \pm \sqrt{70^2 - 4(1)(25)}}{2(1)}

Simplifying the expression under the square root, we get:

x = \frac{-70 \pm \sqrt{4900 - 100}}{2}

x = \frac{-70 \pm \sqrt{4800}}{2}

x = \frac{-70 \pm 69.28}{2}

Now, we have two possible solutions for $x$ :

x_1 = \frac{-70 + 69.28}{2} = -0.36

x_2 = \frac{-70 - 69.28}{2} = -69.64

Alternative Methods

In addition to the quadratic formula, there are other methods for solving quadratic equations, such as factoring and completing the square. However, these methods may not be as straightforward as the quadratic formula, and may require more advanced mathematical techniques.

Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. For example, the equation $x^2 + 70x + 25 = 0$ can be factored as:

(x + 25)(x + 1) = 0

This means that either $(x + 25) = 0$ or $(x + 1) = 0$ . Solving for $x$ , we get:

x + 25 = 0 \Rightarrow x = -25

x + 1 = 0 \Rightarrow x = -1

Completing the Square

Completing the square involves rewriting the quadratic equation in a form that allows us to easily identify the solutions. For example, the equation $x^2 + 70x + 25 = 0$ can be rewritten as:

(x + 35)^2 - 1225 = 0

This means that $(x + 35)^2 = 1225$ . Taking the square root of both sides, we get:

x + 35 = \pm 35

Solving for $x$ , we get:

x + 35 = 35 \Rightarrow x = 0

x + 35 = -35 \Rightarrow x = -70

Conclusion

In this article, we have explored various methods for solving quadratic equations, including the quadratic formula, factoring, and completing the square. We have applied these methods to the given equation, $x^2 + 70x + 25 = 0$ , and obtained two possible solutions for $x$ . We hope that this article has provided a clear and concise guide to solving quadratic equations, and has helped you to understand the different techniques involved.

Final Thoughts

Solving quadratic equations is an essential skill for students and professionals alike. By mastering the different techniques, you can solve a wide range of problems and apply mathematical concepts to real-world situations. Whether you are a student, a teacher, or a professional, we hope that this article has provided you with a valuable resource for solving quadratic equations.

Additional Resources

References

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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, including their definition, properties, and methods for 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, and $a$ cannot be zero.

Q: What are the Properties of a Quadratic Equation?

A: A quadratic equation has several properties, including:

Symmetry: The graph of a quadratic equation is symmetric about its axis of symmetry.
Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex of the parabola.
Vertex: The vertex is the point on the parabola that is farthest from the axis of symmetry.
Intercepts: The x-intercepts are the points on the x-axis where the parabola intersects the x-axis.

Q: How Do I Solve a Quadratic Equation?

A: There are several methods for solving quadratic equations, including:

Quadratic Formula: The quadratic formula is a formula that can be used to solve any quadratic equation. It is given by:

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

Factoring: Factoring involves expressing the quadratic equation as a product of two binomials.
Completing the Square: Completing the square involves rewriting the quadratic equation in a form that allows us to easily identify the solutions.

Q: What is the Quadratic Formula?

A: The quadratic formula is a formula that can be used to solve any quadratic equation. It is given by:

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

This formula can be used to solve any quadratic equation, regardless of the values of $a$ , $b$ , and $c$ .

Q: How Do I Use the Quadratic Formula?

A: To use the quadratic formula, you need to substitute the values of $a$ , $b$ , and $c$ into the formula. For example, if you have the quadratic equation $x^2 + 70x + 25 = 0$ , you can substitute $a = 1$ , $b = 70$ , and $c = 25$ 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. The quadratic formula is a formula that can be used to solve any quadratic equation, while factoring involves expressing the quadratic equation as a product of two binomials.

Q: Can I Use the Quadratic Formula to Solve Any Quadratic Equation?

A: Yes, the quadratic formula can be used to solve any quadratic equation, regardless of the values of $a$ , $b$ , and $c$ .

Q: What is the Axis of Symmetry?

A: The axis of symmetry is the vertical line that passes through the vertex of the parabola.

Q: How Do I Find the Axis of Symmetry?

A: To find the axis of symmetry, you need to find the vertex of the parabola. The vertex is the point on the parabola that is farthest from the axis of symmetry.

Q: What is the Vertex?

A: The vertex is the point on the parabola that is farthest from the axis of symmetry.

Q: How Do I Find the Vertex?

A: To find the vertex, you need to find the x-coordinate of the vertex. The x-coordinate of the vertex is given by:

x = \frac{-b}{2a}

Q: What are the X-Intercepts?

A: The x-intercepts are the points on the x-axis where the parabola intersects the x-axis.

Q: How Do I Find the X-Intercepts?

A: To find the x-intercepts, you need to set the equation equal to zero and solve for $x$ .

Conclusion

In this article, we have addressed some of the most frequently asked questions about quadratic equations, including their definition, properties, and methods for solving them. We hope that this article has provided a clear and concise guide to quadratic equations and has helped you to understand the different techniques involved.