Find The Roots And The Vertex Of The Quadratic On A Calculator. Round All Values To 3 Decimal Places If Necessary.$\[ Y = 4x^2 + 36x - 239 \\]Roots: $ \square $ And $ \square $ Vertex: ($ \square $, $ \square

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will discuss how to find the roots and vertex of a quadratic equation on a calculator, using the given equation $y = 4x^2 + 36x - 239$ as an example.

Understanding the Quadratic Equation

Before we proceed, let's understand the given quadratic equation. The equation is in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, $a = 4$, $b = 36$, and $c = -239$. The roots of the equation are the values of $x$ that satisfy the equation, and the vertex is the maximum or minimum point of the parabola represented by the equation.

Finding the Roots of the Quadratic Equation

To find the roots of the quadratic equation, we can use the quadratic formula:

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

In this case, $a = 4$, $b = 36$, and $c = -239$. Plugging these values into the formula, we get:

$$x = \frac{-36 \pm \sqrt{36^2 - 4(4)(-239)}}{2(4)}$$

Simplifying the expression, we get:

$$x = \frac{-36 \pm \sqrt{1296 + 3824}}{8}$$

$$x = \frac{-36 \pm \sqrt{5120}}{8}$$

$$x = \frac{-36 \pm 71.66}{8}$$

Therefore, the roots of the equation are:

$$x = \frac{-36 + 71.66}{8} = 4.59$$

$$x = \frac{-36 - 71.66}{8} = -12.92$$

Finding the Vertex of the Quadratic Equation

To find the vertex of the quadratic equation, we can use the formula:

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

In this case, $a = 4$ and $b = 36$. Plugging these values into the formula, we get:

$$x = -\frac{36}{2(4)} = -4.5$$

To find the y-coordinate of the vertex, we can plug the value of $x$ into the original equation:

$$y = 4(-4.5)^2 + 36(-4.5) - 239$$

Simplifying the expression, we get:

$$y = 81 - 162 - 239 = -320$$

Therefore, the vertex of the equation is:

$$(x, y) = (-4.5, -320)$$

Using a Calculator to Find the Roots and Vertex

To find the roots and vertex of the quadratic equation on a calculator, we can use the following steps:

1. Enter the equation into the calculator: $y = 4x^2 + 36x - 239$
2. Press the "SOLVE" button to find the roots of the equation.
3. Press the "GRAPH" button to find the vertex of the equation.

Conclusion

In this article, we discussed how to find the roots and vertex of a quadratic equation on a calculator. We used the given equation $y = 4x^2 + 36x - 239$ as an example and showed how to use the quadratic formula and the formula for the vertex to find the roots and vertex of the equation. We also demonstrated how to use a calculator to find the roots and vertex of the equation. By following these steps, you can easily find the roots and vertex of any quadratic equation on a calculator.

Example Problems

1. Find the roots and vertex of the quadratic equation $y = 2x^2 + 5x - 3$.
2. Find the roots and vertex of the quadratic equation $y = x^2 - 4x + 4$.
3. Find the roots and vertex of the quadratic equation $y = 3x^2 + 2x - 1$.

Answer Key

1. Roots: $x = 1.33$ and $x = -2.33$ Vertex: $(x, y) = (-1, -2)$
2. Roots: $x = 2$ and $x = 2$ Vertex: $(x, y) = (2, 0)$
3. Roots: $x = 0.67$ and $x = -1.33$ Vertex: $(x, y) = (-0.5, 2.5)$
   Quadratic Equation Q&A ==========================

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. It is in the form $y = ax^2 + bx + c$, 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 represented by the equation. It is the point where the parabola changes direction.

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

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

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

Q: What is the difference between the roots and the 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 represented by the equation.

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

A: Yes, you can use a calculator to find the roots and vertex of a quadratic equation. Most calculators have a "SOLVE" button that allows you to find the roots of the equation, and a "GRAPH" button that allows you to find the vertex of the equation.

Q: What are some common mistakes to avoid when finding the roots and vertex of a quadratic equation?

A: Some common mistakes to avoid when finding the roots and vertex of a quadratic equation include:

• Not using the correct formula for the roots and vertex
• Not plugging in the correct values for $a$, $b$, and $c$
• Not simplifying the expression correctly
• Not checking for extraneous solutions

Q: Can I use the quadratic formula to find the roots of a quadratic equation with complex roots?

A: Yes, you can use the quadratic formula to find the roots of a quadratic equation with complex roots. The quadratic formula will give you two complex roots, which can be written in the form $x = a + bi$ and $x = a - bi$.

Q: Can I use the quadratic formula to find the roots of a quadratic equation with repeated roots?

A: Yes, you can use the quadratic formula to find the roots of a quadratic equation with repeated roots. The quadratic formula will give you one repeated root, which can be written in the form $x = a$.

Q: Can I use the quadratic formula to find the roots of a quadratic equation with rational roots?

A: Yes, you can use the quadratic formula to find the roots of a quadratic equation with rational roots. The quadratic formula will give you two rational roots, which can be written in the form $x = a/b$ and $x = c/d$.

Q: Can I use the quadratic formula to find the roots of a quadratic equation with irrational roots?

A: Yes, you can use the quadratic formula to find the roots of a quadratic equation with irrational roots. The quadratic formula will give you two irrational roots, which can be written in the form $x = a + \sqrt{b}$ and $x = a - \sqrt{b}$.

Conclusion

In this article, we have discussed some frequently asked questions about quadratic equations, including how to find the roots and vertex of a quadratic equation, how to use a calculator to find the roots and vertex, and some common mistakes to avoid when finding the roots and vertex. We have also discussed some special cases, such as complex roots, repeated roots, rational roots, and irrational roots. By following these tips and techniques, you can easily find the roots and vertex of any quadratic equation.