Find The Roots And The Vertex Of The Quadratic On A Calculator. Round All Values To 3 Decimal Places If Necessary.$y = X^2 + 10x - 39$Answer Attempt 1 Out Of 2:Roots: $\square$ And $\square$Vertex: ($\square$,

Introduction

Quadratic equations are a fundamental concept in mathematics, and finding their roots and vertex is a crucial step in solving various mathematical problems. In this article, we will explore how to find the roots and vertex of a quadratic equation using a calculator. We will use the quadratic equation $y = x^2 + 10x - 39$ as an example.

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 $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In our example, the quadratic equation is $y = x^2 + 10x - 39$, which can be rewritten as $x^2 + 10x - 39 = 0$.

Finding the Roots of a Quadratic Equation

The roots of a quadratic equation are the values of $x$ that satisfy the equation. To find the roots of a quadratic equation, we can use the quadratic formula:

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

In our example, $a = 1$, $b = 10$, and $c = -39$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(-39)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{-10 \pm \sqrt{100 + 156}}{2}$$

$$x = \frac{-10 \pm \sqrt{256}}{2}$$

$$x = \frac{-10 \pm 16}{2}$$

Therefore, the roots of the quadratic equation are:

$$x = \frac{-10 + 16}{2} = 3$$

$$x = \frac{-10 - 16}{2} = -13$$

Finding the Vertex of a Quadratic Equation

The vertex of a quadratic equation is the maximum or minimum point of the parabola. To find the vertex of a quadratic equation, we can use the formula:

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

In our example, $a = 1$ and $b = 10$. Plugging these values into the formula, we get:

$$x = -\frac{10}{2(1)}$$

$$x = -\frac{10}{2}$$

$$x = -5$$

Therefore, the vertex of the quadratic equation is at $x = -5$.

Using a Calculator to Find the Roots and Vertex

To find the roots and vertex of a quadratic equation using a calculator, we can follow these steps:

1. Enter the quadratic equation into the calculator.
2. Use the quadratic formula to find the roots.
3. Use the formula to find the vertex.

For example, let's use a calculator to find the roots and vertex of the quadratic equation $y = x^2 + 10x - 39$.

Step 1: Enter the Quadratic Equation

Enter the quadratic equation $y = x^2 + 10x - 39$ into the calculator.

Step 2: Use the Quadratic Formula to Find the Roots

Use the quadratic formula to find the roots of the quadratic equation.

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

In this case, $a = 1$, $b = 10$, and $c = -39$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(-39)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{-10 \pm \sqrt{100 + 156}}{2}$$

$$x = \frac{-10 \pm \sqrt{256}}{2}$$

$$x = \frac{-10 \pm 16}{2}$$

Therefore, the roots of the quadratic equation are:

$$x = \frac{-10 + 16}{2} = 3$$

$$x = \frac{-10 - 16}{2} = -13$$

Step 3: Use the Formula to Find the Vertex

Use the formula to find the vertex of the quadratic equation.

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

In this case, $a = 1$ and $b = 10$. Plugging these values into the formula, we get:

$$x = -\frac{10}{2(1)}$$

$$x = -\frac{10}{2}$$

$$x = -5$$

Therefore, the vertex of the quadratic equation is at $x = -5$.

Conclusion

In this article, we have explored how to find the roots and vertex of a quadratic equation using a calculator. We have used the quadratic formula to find the roots and the formula to find the vertex. We have also provided a step-by-step guide on how to use a calculator to find the roots and vertex of a quadratic equation. By following these steps, you can easily find the roots and vertex of a quadratic equation using a calculator.

Final Answer

The final answer is:

$$$$ $$

Frequently Asked Questions

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. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

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

A: To find the roots of a quadratic equation, you can use the quadratic formula:

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

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the maximum or minimum point of the parabola. To find the vertex of a quadratic equation, you can use the formula:

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

Q: How do I use a calculator to find the roots and vertex of a quadratic equation?

A: To use a calculator to find the roots and vertex of a quadratic equation, follow these steps:

1. Enter the quadratic equation into the calculator.
2. Use the quadratic formula to find the roots.
3. Use the formula to find the vertex.

Q: What is the difference between the roots and vertex of a quadratic equation?

A: The roots of a quadratic equation are the values of $x$ that satisfy the equation, while the vertex is the maximum or minimum point of the parabola.

Q: Can a quadratic equation have more than two roots?

A: No, a quadratic equation can only have two roots.

Q: Can a quadratic equation have no roots?

A: Yes, a quadratic equation can have no roots if the discriminant ($b^2 - 4ac$) is negative.

Q: Can a quadratic equation have a root that is a complex number?

A: Yes, a quadratic equation can have a root that is a complex number if the discriminant ($b^2 - 4ac$) is negative.

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 the points on a coordinate plane.

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

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

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

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

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

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic equations. We hope that this article has provided you with a better understanding of quadratic equations and how to work with them.

Final Answer

The final answer is:

• Roots: $3$ and $-13$
• Vertex: $(-5, 0)$
• Axis of symmetry: $x = -5$