Question

Jamal
Step 1: Set equation =0 . x^2+4x−12=0
Step 2: Write quadratic equation as the product of two factors. (x+6)(x−2)=0
Step 3: Set each factor equal to 0. x+6=0 and x−2=0
Step 4: Solve each equation. x=−6 and x=2 George
Step 1: Begin with the equation. x^2+4x−5=7
Step 2: Write quadratic equation as the product of two factors. (x+5)(x−1)=7
Step 3: Set each factor equal to 0. x+5=0 and x−1=0
Step 4: Solve each equation. x=−5 and x=1
When trying to solve the quadratic equation x^2+4x−5=7 , Jamal and George each provided their work in the tables. Each said they used the Zero Product Property to solve the equation after step 2. Explain how you know the correct student’s solution is accurate and how you know the incorrect student’s solution is inaccurate. simple explanation

Answer 1

Determine the zero(s) of the quadratic function y=6x^2+13x+5 . Show all steps. Simple explanation

Answer 2

shorten the explanation,

Answer 3

Complete the square to determine the vertex and axis of symmetry of y=−x^2+8x+7 .

short and simple explanation

Answer 4

To determine which student's solution is accurate, we can look at their steps and where they diverged in their methods.

**Jamal's Steps:**
1. He started with the equation \( x^2 + 4x - 12 = 0 \). This is a proper quadratic equation set to zero, which is correct.
2. He factored it as \( (x + 6)(x - 2) = 0 \).
3. He set \( x + 6 = 0 \) and \( x - 2 = 0 \).
4. He solved for \( x \) and got \( x = -6 \) and \( x = 2 \).

However, Jamal's original equation was incorrect. \( x^2 + 4x - 12 = 0 \) is not the same as \( x^2 + 4x - 5 = 7 \). Thus, his solution is inaccurate.

**George's Steps:**
1. George began with the equation \( x^2 + 4x - 5 = 7 \).
2. He should have set the equation to zero first: \( x^2 + 4x - 5 - 7 = 0\) or \( x^2 + 4x - 12 = 0\).
3. However, he incorrectly factored \( (x+5)(x-1) = 7 \), which does not correspond to either side of the equation.
4. He then attempted to use the Zero Product Property incorrectly and set factors equal to zero based on incorrect factoring.

Both Jamal and George made errors in their work. However, Jamal at least started with an equation that could be factored (even though it was not the right one). George misinterpreted his own equation and derived factors that did not align with the original equation.

In conclusion, both students made mistakes: Jamal did not start with the correct equation, and George improperly applied the Zero Product Property to incorrect factors. Neither solution is accurate, but George's misunderstanding of the factors shows a greater error in his approach to solving the original equation.

Answer 5

To determine the zero(s) of the quadratic function \( y = 6x^2 + 13x + 5 \), we can follow these steps:

### Step 1: Set the Equation to Zero
Set the quadratic function equal to zero to find the x-values where the function intersects the x-axis.

\[
6x^2 + 13x + 5 = 0
\]

### Step 2: Factor the Quadratic
Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to \( 6 \times 5 = 30 \) (the product of the coefficient of \( x^2 \) and the constant term) and add to \( 13 \) (the coefficient of \( x \)).

These two numbers are \( 3 \) and \( 10 \) because:
- \( 3 \times 10 = 30 \)
- \( 3 + 10 = 13 \)

### Step 3: Rewrite the Middle Term
We can rewrite the quadratic based on those factors:

\[
6x^2 + 3x + 10x + 5 = 0
\]

### Step 4: Group Terms
Now, group the terms:

\[
(6x^2 + 3x) + (10x + 5) = 0
\]

### Step 5: Factor by Grouping
Factor out the common factors in each group:

\[
3x(2x + 1) + 5(2x + 1) = 0
\]

Now, factor out \( (2x + 1) \):

\[
(2x + 1)(3x + 5) = 0
\]

### Step 6: Set Each Factor to Zero
Now, we can apply the Zero Product Property. Set each factor equal to zero:

1. \( 2x + 1 = 0 \)
2. \( 3x + 5 = 0 \)

### Step 7: Solve Each Equation
**For the first factor:**

\[
2x + 1 = 0 \implies 2x = -1 \implies x = -\frac{1}{2}
\]

**For the second factor:**

\[
3x + 5 = 0 \implies 3x = -5 \implies x = -\frac{5}{3}
\]

### Step 8: State the Zeros
The zeros of the quadratic function \( y = 6x^2 + 13x + 5 \) are:

\[
x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{5}{3}
\]

### Conclusion
The values \( x = -\frac{1}{2} \) and \( x = -\frac{5}{3} \) are the points where the graph of the quadratic function intersects the x-axis.

Answer 6

To find the zeros of the quadratic function \( y = 6x^2 + 13x + 5 \):

### Step 1: Set the Equation to Zero
\[
6x^2 + 13x + 5 = 0
\]

### Step 2: Factor the Quadratic
Look for two numbers that multiply to \( 30 \) (from \( 6 \times 5 \)) and add to \( 13 \):
- The numbers are \( 3 \) and \( 10 \).

Rewrite the equation:
\[
6x^2 + 3x + 10x + 5 = 0
\]

### Step 3: Group and Factor
Group the terms:
\[
(6x^2 + 3x) + (10x + 5) = 0
\]

Factor out common terms:
\[
3x(2x + 1) + 5(2x + 1) = 0
\]
\[
(2x + 1)(3x + 5) = 0
\]

### Step 4: Solve Each Factor
Set each factor to zero:
1. \( 2x + 1 = 0 \) ⇒ \( x = -\frac{1}{2} \)
2. \( 3x + 5 = 0 \) ⇒ \( x = -\frac{5}{3} \)

### Conclusion
The zeros of the function are:
\[
x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{5}{3}
\]

Answer 7

To complete the square for the function \( y = -x^2 + 8x + 7 \):

### Step 1: Factor out the coefficient of \( x^2 \)
Factor out \(-1\) from the quadratic terms:
\[
y = -(x^2 - 8x) + 7
\]

### Step 2: Complete the square
To complete the square, take half of the coefficient of \( x \) (which is \(-8\)), square it, and add it inside the parentheses:
- Half of \(-8\) is \(-4\) and \((-4)^2 = 16\).

Now adjust the equation:
\[
y = -\left(x^2 - 8x + 16 - 16\right) + 7
\]
\[
y = -\left((x - 4)^2 - 16\right) + 7
\]
\[
y = -(x - 4)^2 + 16 + 7
\]
\[
y = -(x - 4)^2 + 23
\]

### Step 3: Identify the vertex and axis of symmetry
The vertex form of the equation is \( y = -(x - 4)^2 + 23 \).

- **Vertex**: \( (4, 23) \)
- **Axis of Symmetry**: \( x = 4 \)

### Conclusion
The vertex is \( (4, 23) \) and the axis of symmetry is \( x = 4 \).