Solve The Quadratic Equation. Separate Solutions With A Comma.$x^2 + 11x - 60 = 0$x =$

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 quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. We will use the quadratic formula to solve the equation $x^2 + 11x - 60 = 0$.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The graph of a quadratic equation is a parabola, which is a U-shaped curve.

The Quadratic Formula

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

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

where $a$, $b$, and $c$ are the constants in the quadratic equation.

Solving the Equation $x^2 + 11x - 60 = 0$

To solve the equation $x^2 + 11x - 60 = 0$, we can use the quadratic formula. We have $a = 1$, $b = 11$, and $c = -60$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-11 \pm \sqrt{11^2 - 4(1)(-60)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{-11 \pm \sqrt{121 + 240}}{2}$$

$$x = \frac{-11 \pm \sqrt{361}}{2}$$

$$x = \frac{-11 \pm 19}{2}$$

Therefore, the solutions to the equation $x^2 + 11x - 60 = 0$ are:

$$x = \frac{-11 + 19}{2} = 4$$

$$x = \frac{-11 - 19}{2} = -15$$

Conclusion

Solving quadratic equations is an essential skill in mathematics, and the quadratic formula is a powerful tool for solving these equations. By following the steps outlined in this article, we can solve quadratic equations of the form $ax^2 + bx + c = 0$. In this article, we solved the equation $x^2 + 11x - 60 = 0$ using the quadratic formula.

Tips and Tricks

• Make sure to plug in the correct values for $a$, $b$, and $c$ into the quadratic formula.
• Simplify the expression under the square root before solving for $x$.
• Check your solutions by plugging them back into the original equation.

Real-World Applications

Quadratic equations have many real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Common Mistakes

• Not simplifying the expression under the square root: Failing to simplify the expression under the square root can lead to incorrect solutions.
• Not checking solutions: Failing to check solutions by plugging them back into the original equation can lead to incorrect answers.

Conclusion

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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 answer some of the most frequently asked questions about 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. It is typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where $a$, $b$, and $c$ are the constants in the quadratic equation.

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

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression under the square root and solve for $x$.

Q: What if the expression under the square root is negative?

A: If the expression under the square root is negative, then the quadratic equation has no real solutions. In this case, the solutions are complex numbers.

Q: How do I check my solutions?

A: To check your solutions, plug them back into the original equation. If the solutions satisfy the equation, then they are correct.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Not simplifying the expression under the square root
• Not checking solutions
• Not using the correct values for $a$, $b$, and $c$

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use a graphing calculator or a computer program. You can also use a table of values to plot points on the graph.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph where the parabola changes direction. It is given by the formula:

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

Q: How do I find the x-intercepts of a quadratic equation?

A: To find the x-intercepts of a quadratic equation, you can set the equation equal to zero and solve for $x$. The x-intercepts are the points where the graph crosses the x-axis.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we answered some of the most frequently asked questions about quadratic equations. We hope this article has been helpful in clarifying any doubts you may have had about quadratic equations.

Additional Resources

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equations
• Wolfram Alpha: Quadratic Equations

Common Quadratic Equation Problems

• $x^2 + 5x + 6 = 0$
• $x^2 - 7x + 12 = 0$
• $x^2 + 2x - 15 = 0$

Solving Quadratic Equations: A Step-by-Step Guide

• Step 1: Write the quadratic equation in the form $ax^2 + bx + c = 0$
• Step 2: Plug in the values of $a$, $b$, and $c$ into the quadratic formula
• Step 3: Simplify the expression under the square root
• Step 4: Solve for $x$

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we answered some of the most frequently asked questions about quadratic equations. We hope this article has been helpful in clarifying any doubts you may have had about quadratic equations.