Find The Roots And The Vertex Of The Quadratic Using A Calculator. Round All Values To Three Decimal Places If Necessary.$\[ Y = 4x^2 + 36x - 239 \\]Answer Attempt 1 Out Of 2:- Roots: \[$\square\$\] And \[$\square\$\]- Vertex:

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding how to find their roots and vertex is crucial for solving various mathematical problems. In this article, we will explore how to use a calculator to find the roots and vertex of a quadratic equation. We will use the quadratic equation $y = 4x^2 + 36x - 239$ as an example.

Using a Calculator to Find the Roots

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

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. In our example, $a = 4$, $b = 36$, and $c = -239$.

We can plug these values into the quadratic formula to find the roots:

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

Using a calculator, we can evaluate the expression under the square root:

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

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

$$x = \frac{-36 \pm 71.677}{8}$$

Therefore, the roots of the quadratic equation are:

$$x = \frac{-36 + 71.677}{8} = 6.336$$

$$x = \frac{-36 - 71.677}{8} = -14.592$$

Rounding Values to Three Decimal Places

Since we are asked to round all values to three decimal places, we can round the roots to three decimal places:

$$x = 6.336$$

$$x = -14.592$$

Finding the Vertex

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

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

Using the values of $a$ and $b$ from our quadratic equation, we can plug them into the formula:

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

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

$$x = -4.5$$

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

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

$$y = 81 - 162 - 239$$

$$y = -320$$

Therefore, the vertex of the quadratic equation is:

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

Conclusion

In this article, we have shown how to use a calculator to find the roots and vertex of a quadratic equation. We have used the quadratic formula to find the roots and the formula for the x-coordinate of the vertex to find the vertex. We have also rounded all values to three decimal places as requested. This article provides a step-by-step guide on how to find the roots and vertex of a quadratic equation using a calculator.

Example Problems

1. Find the roots and vertex of the quadratic equation $y = 2x^2 + 12x - 15$.
2. Find the roots and vertex of the quadratic equation $y = 3x^2 - 6x + 2$.

Answer Key

1. Roots: $x = 3.5$ and $x = -2.5$ Vertex: $(x, y) = (-1, -10)$
2. Roots: $x = 1$ and $x = -2$ Vertex: $(x, y) = (1, 0)$
   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 (usually x) 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 find the roots of a quadratic equation?

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

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

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 typically denoted as $(h, k)$, where $h$ is the x-coordinate and $k$ is the y-coordinate.

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:

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

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

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

A: The roots of a quadratic equation are the points where the graph intersects the x-axis, while the vertex is the point on the graph where the parabola changes direction.

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. Simply plug in the values of $a$, $b$, and $c$ into the quadratic formula or the formula for the vertex.

Q: What is the significance of the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It allows you to find the roots of a quadratic equation without having to factor the equation or use other methods.

Q: Can I use the quadratic formula to solve quadratic equations with complex roots?

A: Yes, you can use the quadratic formula to solve quadratic equations with complex roots. The formula will give you the complex roots of the equation.

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

A: The roots of a quadratic equation are symmetric about the vertex. This means that if you know the vertex, you can find the roots by using the formula:

$$x = h \pm \sqrt{(k - f(h))^2 + (g(h) - k)^2}$$

where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate.

Q: Can I use the quadratic formula to solve quadratic equations with rational roots?

A: Yes, you can use the quadratic formula to solve quadratic equations with rational roots. The formula will give you the rational roots of the equation.

Q: What is the significance of the vertex of a quadratic equation?

A: The vertex of a quadratic equation is an important concept in mathematics. It represents the point on the graph where the parabola changes direction, and it is used in a variety of applications, including physics, engineering, and economics.

Q: Can I use the quadratic formula to solve quadratic equations with irrational roots?

A: Yes, you can use the quadratic formula to solve quadratic equations with irrational roots. The formula will give you the irrational roots of the equation.

Q: What is the relationship between the quadratic formula and the graph of a quadratic equation?

A: The quadratic formula is closely related to the graph of a quadratic equation. The formula gives you the x-coordinates of the roots, which are the points where the graph intersects the x-axis. The graph of a quadratic equation is a parabola that opens upward or downward, and the vertex is the point on the graph where the parabola changes direction.