iven the equation below, determine the x-intercept

﻿y space equals space 4 x squared minus 11 x minus 3﻿


﻿x space equals space minus 0.63 space o r space 4.32﻿


﻿x space equals space 0.72 space o r space 5.53﻿


﻿x space equals space minus 3.42 space o r space 0.23﻿


None of the Above

Question 2
1
 Point
Given the equation below, determine the y-intercept

﻿y space equals space 4 x squared minus 11 x minus 3﻿


﻿negative 11﻿


﻿11﻿


4


﻿negative 3﻿

Question 3
1
 Point
Given the equation below, determine the y coordinate of the vertex

﻿y space equals space left parenthesis 2 x minus 4 right parenthesis left parenthesis 2 minus 4 x right parenthesis﻿


﻿2.5﻿


﻿4.5﻿


﻿negative 3﻿


﻿9﻿

Question 4
1
 Point
Given the equation below, determine the x-coordinate of the vertex

﻿y space equals space 5 x squared minus 7 x minus 24﻿


﻿0.7﻿


﻿3﻿


﻿negative 26.45﻿


﻿negative 1.6﻿

Question 5
1
 Point
Given the equation below, determine the Vertex

﻿y space equals space minus 2.5 x squared plus 7 x minus 5﻿


(-2.0, 1.0)


(-0.4, 0.7)


(1.7, -0.3)


(1.4, -0.1)



Let's solve each question one by one.

### Question 1: Finding the x-intercept
The x-intercept occurs when \( y = 0 \). Given the quadratic equation:

\[
y = 4x^2 - 11x - 3
\]

To find the x-intercepts, we need to solve for \( x \) by setting \( y = 0 \):

\[
0 = 4x^2 - 11x - 3
\]

We can solve this quadratic equation using the quadratic formula:

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

Where \( a = 4 \), \( b = -11 \), and \( c = -3 \):

\[
x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(4)(-3)}}{2(4)}
\]

Solving this will give the correct x-intercepts. However, based on the provided choices, none match the exact roots for this equation. Therefore, the correct answer is **None of the Above**.

---

### Question 2: Finding the y-intercept
The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) in the given equation:

\[
y = 4x^2 - 11x - 3
\]

Substitute \( x = 0 \):

\[
y = 4(0)^2 - 11(0) - 3 = -3
\]

So, the y-intercept is **-3**.

---

### Question 3: Finding the y-coordinate of the vertex
The equation is:

\[
y = (2x - 4)(2 - 4x)
\]

We can expand this expression:

\[
y = (2x)(2) + (2x)(-4x) + (-4)(2) + (-4)(-4x)
\]
\[
y = 4x - 8x^2 - 8 + 16x
\]
\[
y = -8x^2 + 20x - 8
\]

The y-coordinate of the vertex can be found using the formula \( y = \frac{-D}{4a} \), where \( D \) is the discriminant. Alternatively, find the x-coordinate and substitute it back into the equation. Upon solving, the y-coordinate is **9**.

---

### Question 4: Finding the x-coordinate of the vertex
The given equation is:

\[
y = 5x^2 - 7x - 24
\]

The x-coordinate of the vertex can be found using the formula \( x = \frac{-b}{2a} \). Here, \( a = 5 \), \( b = -7 \):

\[
x = \frac{-(-7)}{2(5)} = \frac{7}{10} = 0.7
\]

So, the x-coordinate of the vertex is **0.7**.

---

### Question 5: Finding the vertex
The equation is:

\[
y = -2.5x^2 + 7x - 5
\]

The x-coordinate of the vertex is given by \( x = \frac{-b}{2a} \), where \( a = -2.5 \) and \( b = 7 \):

\[
x = \frac{-7}{2(-2.5)} = 1.4
\]

Now, substitute \( x = 1.4 \) back into the equation to find the y-coordinate:

\[
y = -2.5(1.4)^2 + 7(1.4) - 5
\]
\[
y = -2.5(1.96) + 9.8 - 5 = -4.9 + 9.8 - 5 = -0.1
\]

Thus, the vertex is \( (1.4, -0.1) \).

---

Do you want more details on any question?  
Here are 5 related questions:
1. How do you derive the quadratic formula for solving equations?
2. What are the properties of a parabola's vertex and intercepts?
3. How do you determine the axis of symmetry in a quadratic equation?
4. How can completing the square help in finding the vertex form of a quadratic equation?
5. What are the discriminant's implications for the nature of quadratic roots?

**Tip**: When solving for x-intercepts of quadratic equations, always use the quadratic formula if factoring is not easy.

Given the equation below, determine the x-intercept

y = 4x^2 - 11x - 3

x = -0.63 or 4.32

x = 0.72 or 5.53

x = -3.42 or 0.23

None of the Above

Given the equation below, determine the y-intercept

y = 4x^2 - 11x - 3

-11

11

4

-3

Given the equation below, determine the y-coordinate of the vertex

y = (2x - 4)(2 - 4x)

2.5

4.5

-3

9

Given the equation below, determine the x-coordinate of the vertex

y = 5x^2 - 7x - 24

0.7

3

-26.45

-1.6

Given the equation below, determine the Vertex

y = -2.5x^2 + 7x - 5

(-2.0, 1.0)

(-0.4, 0.7)

(1.7, -0.3)

(1.4, -0.1)

Learn how to solve quadratic equations, find x- and y-intercepts, and determine the vertex. This problem involves key algebraic concepts and is suitable for students in Grades 9-10. Detailed step-by-step solutions are provided for equations such as y = 4x^2 - 11x - 3 and y = -2.5x^2 + 7x - 5.

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