Solve For $x$.$0 = X^2 + 8x + 15$Enter Your Answers In The Boxes.The Solutions Are $ □ \square □ [/tex] And $\square$.

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Introduction

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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 find the solutions to the equation $0 = x^2 + 8x + 15$.

What is a Quadratic Equation?

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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

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

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

This formula will give us two solutions for the equation $ax^2 + bx + c = 0$.

Solving the Equation $0 = x^2 + 8x + 15$

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Now, let's apply the quadratic formula to the equation $0 = x^2 + 8x + 15$. We have:

$a = 1$, $b = 8$, and $c = 15$

Plugging these values into the quadratic formula, we get:

$$x = \frac{-8 \pm \sqrt{8^2 - 4(1)(15)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{-8 \pm \sqrt{64 - 60}}{2}$$

$$x = \frac{-8 \pm \sqrt{4}}{2}$$

$$x = \frac{-8 \pm 2}{2}$$

This gives us two possible solutions:

$$x = \frac{-8 + 2}{2} = -3$$

$$x = \frac{-8 - 2}{2} = -5$$

Checking the Solutions

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To check our solutions, we can plug them back into the original equation:

$$0 = (-3)^2 + 8(-3) + 15$$

$$0 = 9 - 24 + 15$$

$$0 = 0$$

This confirms that $x = -3$ is a solution to the equation.

Similarly, we can check the second solution:

$$0 = (-5)^2 + 8(-5) + 15$$

$$0 = 25 - 40 + 15$$

$$0 = 0$$

This confirms that $x = -5$ is also a solution to the equation.

Conclusion

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In this article, we have solved the quadratic equation $0 = x^2 + 8x + 15$ using the quadratic formula. We have found two solutions, $x = -3$ and $x = -5$, and checked them to confirm that they are indeed solutions to the equation. The quadratic formula is a powerful tool for solving quadratic equations, and it is an essential concept in mathematics.

Tips and Tricks

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• When using the quadratic formula, make sure to plug in the correct values for $a$, $b$, and $c$.
• 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

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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 the behavior of economic systems, such as supply and demand.

Final Thoughts

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Solving quadratic equations is an essential skill for students and professionals alike. The quadratic formula is a powerful tool for solving quadratic equations, and it is an essential concept in mathematics. By following the steps outlined in this article, you can solve quadratic equations with ease and confidence.

Additional Resources

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

Frequently Asked Questions

• What is a quadratic equation?
• How do I solve a quadratic equation?
• What is the quadratic formula?
• How do I check my solutions?

Glossary

• Quadratic equation: A polynomial equation of degree two.
• Quadratic formula: A formula for solving quadratic equations.
• Parabola: A U-shaped curve that is the graph of a quadratic equation.
• Solution: A value that satisfies a quadratic equation.
  

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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?

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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: How do I solve a quadratic equation?

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A: There are several methods for solving quadratic equations, including factoring, using the quadratic formula, and graphing. The quadratic formula is a powerful tool for solving quadratic equations and is given by:

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

Q: What is the quadratic formula?

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

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

Q: How do I check my solutions?

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A: To check your solutions, plug them back into the original equation. If the equation is true, then the solution is correct.

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

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A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a parabolic graph, while a linear equation has a straight line graph.

Q: Can I use the quadratic formula to solve any quadratic equation?

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A: Yes, the quadratic formula can be used to solve any quadratic equation. However, it is not always the easiest method to use, and other methods such as factoring or graphing may be more efficient.

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

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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 the behavior of economic systems, such as supply and demand.

Q: How do I graph a quadratic equation?

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

Q: What is the vertex of a quadratic equation?

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A: The vertex of a quadratic equation is the point on the graph where the parabola changes direction. It is the minimum or maximum point on the graph.

Q: How do I find the vertex of a quadratic equation?

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A: To find the vertex of a quadratic equation, use the formula:

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

This will give you the x-coordinate of the vertex. To find the y-coordinate, plug the x-coordinate back into the original equation.

Q: What is the axis of symmetry of a quadratic equation?

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A: The axis of symmetry of a quadratic equation is a vertical line that passes through the vertex of the graph. It is the line of symmetry for the parabola.

Q: How do I find the axis of symmetry of a quadratic equation?

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A: To find the axis of symmetry of a quadratic equation, use the formula:

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

This will give you the x-coordinate of the axis of symmetry.

Conclusion

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Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have answered some of the most frequently asked questions about quadratic equations. We hope that this article has been helpful in understanding quadratic equations and how to solve them.

Additional Resources

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

Glossary

• Quadratic equation: A polynomial equation of degree two.
• Quadratic formula: A formula for solving quadratic equations.
• Parabola: A U-shaped curve that is the graph of a quadratic equation.
• Solution: A value that satisfies a quadratic equation.
• Vertex: The point on the graph where the parabola changes direction.
• Axis of symmetry: A vertical line that passes through the vertex of the graph.
• Quadratic equation: A polynomial equation of degree two.

Frequently Asked Questions

• What is a quadratic equation?
• How do I solve a quadratic equation?
• What is the quadratic formula?
• How do I check my solutions?
• What is the difference between a quadratic equation and a linear equation?
• Can I use the quadratic formula to solve any quadratic equation?
• What are some real-world applications of quadratic equations?
• How do I graph a quadratic equation?
• What is the vertex of a quadratic equation?
• How do I find the vertex of a quadratic equation?
• What is the axis of symmetry of a quadratic equation?
• How do I find the axis of symmetry of a quadratic equation?