Question

edida solved the quadratic equation x2−96=4x
by factoring. Her work is shown below. At which step did Jedida first make a mistake, if at all?

x2−96=4x
Step 1: x2+4x−96=0
Add 4x
to both sides.
Step 2: (x+12)(x−8)=0
Factor.
Step 3: x+12=0
or x−8=0
Use the Zero Product Property.
x=−12
or x=8
(1 point)
Responses

Step 3; she did not apply the Zero Product Property correctly.
Step 3; she did not apply the Zero Product Property correctly.

She did not make any mistakes.
She did not make any mistakes.

Step 2; she did not factor the left side of the equation correctly.
Step 2; she did not factor the left side of the equation correctly.

Step 1; she added 4x
to both sides instead of subtracting.
Step 1; she added 4 x to both sides instead of subtracting.
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Answer 1

Let's analyze Jedida's work step by step:

1. Starting with the equation:
\( x^2 - 96 = 4x \)

2. In Step 1, she rewrites the equation as:
\( x^2 + 4x - 96 = 0 \)
Here, she correctly added \( 4x \) to both sides.

3. In Step 2, she factored the left side as:
\( (x + 12)(x - 8) = 0 \)
To check this factorization, we can expand it:
\[
(x + 12)(x - 8) = x^2 - 8x + 12x - 96 = x^2 + 4x - 96
\]
Thus, this factorization is also correct.

4. In Step 3, she applies the Zero Product Property and sets each factor to zero:
\( x + 12 = 0 \) or \( x - 8 = 0 \)

5. Solving these gives:
\( x = -12 \) or \( x = 8 \)

Jedida did not make any mistakes in her calculations. All the steps are correct as shown above.

Therefore, the response is:
**She did not make any mistakes.**

Answer 2

Solve the following quadratic equation using the perfect square trinomial pattern: x2+x+0.25=0
.(1 point)
Responses

x=−0.25
x equals negative 0.25

x=−0.5
x equals negative 0.5

x=0.5
x equals 0.5

x=0.25

Answer 3

To solve the quadratic equation \( x^2 + x + 0.25 = 0 \) using the perfect square trinomial pattern, we can rewrite it in a way that shows it is a perfect square.

The equation can be expressed as:

\[
x^2 + x + \left(\frac{1}{2}\right)^2 = 0
\]

This is because \( 0.25 = \left(\frac{1}{2}\right)^2 \).

Now we can factor the left side:

\[
(x + 0.5)^2 = 0
\]

Next, we take the square root of both sides:

\[
x + 0.5 = 0
\]

Now, we solve for \( x \):

\[
x = -0.5
\]

So the solution to the equation \( x^2 + x + 0.25 = 0 \) is:

**x = -0.5**.

Answer 4

Ginger was asked to solve 25x2−196=0
by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point)
Responses

There is no solution for x because the difference of squares pattern does not apply.
There is no solution for x because the difference of squares pattern does not apply.

x=19625
x equals Start Fraction 196 over 25 End Fraction

x=−145
, x=145
x equals negative Start Fraction 14 over 5 End Fraction , x equals Start Fraction 14 over 5 End Fraction

x=145

Answer 5

To solve the equation \( 25x^2 - 196 = 0 \) using the difference of squares pattern, we can recognize that it can be rewritten as:

\[
(5x)^2 - 14^2 = 0
\]

This matches the form of a difference of squares, which states that \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = 5x \) and \( b = 14 \).

Now, we can factor the equation:

\[
(5x - 14)(5x + 14) = 0
\]

Next, we apply the Zero Product Property, which tells us to set each factor equal to zero:

1. \( 5x - 14 = 0 \)
- Solving for \( x \):
\[
5x = 14 \quad \Rightarrow \quad x = \frac{14}{5}
\]

2. \( 5x + 14 = 0 \)
- Solving for \( x \):
\[
5x = -14 \quad \Rightarrow \quad x = -\frac{14}{5}
\]

So, the complete solution set is:

\[
x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5}
\]

Thus, the correct response is:

**x = -14/5, x = 14/5.**

Answer 6

Complete the square to solve the following quadratic equation: x2−6x−20=20
.(1 point)
Responses

x=52
, x=−46
x equals 52 , x equals negative 46

x=10
, x=−4
x equals 10 , x equals negative 4

x=−10
, x=4
x equals negative 10 , x equals 4

x=3+210−−√, 3−210−−√
x equals 3 plus 2 Start Root 10 End Root comma 3 minus 2 Start Root 10 End Root

Answer 7

To solve the quadratic equation \( x^2 - 6x - 20 = 20 \) by completing the square, we can follow these steps:

1. **Rewrite the equation:**
Move everything to one side:
\[
x^2 - 6x - 20 - 20 = 0
\]
Simplifying this gives:
\[
x^2 - 6x - 40 = 0
\]

2. **Move the constant to the other side:**
\[
x^2 - 6x = 40
\]

3. **Complete the square:**
To complete the square, take half of the coefficient of \( x \) (which is -6), square it, and then add it to both sides. Half of -6 is -3, and squaring it gives 9.
\[
x^2 - 6x + 9 = 40 + 9
\]
This simplifies to:
\[
(x - 3)^2 = 49
\]

4. **Take the square root of both sides:**
\[
x - 3 = \pm 7
\]

5. **Solve for \( x \):**
This produces two equations:
\[
x - 3 = 7 \quad \Rightarrow \quad x = 10
\]
\[
x - 3 = -7 \quad \Rightarrow \quad x = -4
\]

Thus, the solutions to the equation are:
\[
x = 10 \quad \text{and} \quad x = -4
\]

So the correct response is:
**x = 10, x = -4.**

Answer 8

Using the quadratic formula, find the solution to 3x2+18x+24=0
.(1 point)
Responses

x=−2
x equals negative 2

x=−2
, x=−4
x equals negative 2 , x equals negative 4

x=4
, x=2
x equals 4 , x equals 2

x=−4

Answer 9

To solve the quadratic equation \( 3x^2 + 18x + 24 = 0 \) using the quadratic formula, we will first identify the coefficients \( a \), \( b \), and \( c \):

- \( a = 3 \)
- \( b = 18 \)
- \( c = 24 \)

The quadratic formula is given by:

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

Now we can substitute the coefficients into the formula:

1. **Calculate the discriminant**:
\[
b^2 - 4ac = (18)^2 - 4(3)(24)
\]
\[
= 324 - 288
\]
\[
= 36
\]

2. **Apply the quadratic formula**:
\[
x = \frac{{-18 \pm \sqrt{36}}}{2(3)}
\]
\[
= \frac{{-18 \pm 6}}{6}
\]

3. **Calculate the two possible solutions**:
- For the plus case:
\[
x = \frac{{-18 + 6}}{6} = \frac{{-12}}{6} = -2
\]
- For the minus case:
\[
x = \frac{{-18 - 6}}{6} = \frac{{-24}}{6} = -4
\]

Thus, the solutions are:
\[
x = -2 \quad \text{and} \quad x = -4
\]

Therefore, the correct response is:
**x = -2, x = -4.**