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)

Question

Responses

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

x=145
x equals Start Fraction 14 over 5 End Fraction

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=−145, x=145

Answer 1

To solve the equation \( 25x^2 - 196 = 0 \) by factoring using the difference of squares pattern, we start by recognizing that this equation can be expressed in the form \( a^2 - b^2 = 0 \), where:

• \( a^2 = 25x^2 \) (which means \( a = 5x \))
• \( b^2 = 196 \) (which means \( b = 14 \))

Thus, we can rewrite the equation as:

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

Using the difference of squares factorization, it can be factored to:

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

Now, we set each factor equal to zero:

1. \( 5x - 14 = 0 \)

   • \( 5x = 14 \)
   • \( x = \frac{14}{5} \)

2. \( 5x + 14 = 0 \)

   • \( 5x = -14 \)
   • \( x = -\frac{14}{5} \)

So the solutions are:

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

Therefore, the correct response is:

x = −145, x = 145 (This indicates \( x = -\frac{14}{5} \) and \( x = \frac{14}{5} \)) - note that the representation \( 145 \) is likely a mistake since it should read as \( \frac{14}{5} \).

Answer 2

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

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

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

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

x=10, x=−4

Answer 3

Let's start by solving the quadratic equation \( x^2 - 6x - 20 = 20 \) by completing the square.

1. Rearrange the equation:

   \[ x^2 - 6x - 20 - 20 = 0 \]

   Simplifying gives:

   \[ x^2 - 6x - 40 = 0 \]

2. Move the constant to the right:

   \[ x^2 - 6x = 40 \]

3. Complete the square:

   • Take half of the coefficient of \( x \) (which is \(-6\)), square it, and add it to both sides.
   • Half of \(-6\) is \(-3\), and \((-3)^2 = 9\).

   Adding 9 to both sides gives:

   \[ x^2 - 6x + 9 = 40 + 9 \]

   This simplifies to:

   \[ (x - 3)^2 = 49 \]

4. Take the square root of both sides:

   \[ x - 3 = \pm 7 \]

5. Solve for \( x \):

   • For \( x - 3 = 7 \): \[ x = 10 \]

   • For \( x - 3 = -7 \): \[ x = -4 \]

Thus, the solutions to the equation are:

\[ x = 10, \quad x = -4 \]

Therefore, the correct response is:

x = 10, x = −4

Answer 4

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 equals negative 4

x=4, x=2

Answer 5

To find the solutions to the quadratic equation \( 3x^2 + 18x + 24 = 0 \) using the quadratic formula, we start by identifying the coefficients:

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

The quadratic formula is given by:

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

Next, we calculate the discriminant \( b^2 - 4ac \):

\[ b^2 = 18^2 = 324 \] \[ 4ac = 4 \cdot 3 \cdot 24 = 288 \] \[ \text{Discriminant} = 324 - 288 = 36 \]

Now, we can use the quadratic formula:

\[ x = \frac{-18 \pm \sqrt{36}}{2 \cdot 3} \] \[ x = \frac{-18 \pm 6}{6} \]

This gives us two possible solutions:

1. First solution:

   \[ x = \frac{-18 + 6}{6} = \frac{-12}{6} = -2 \]

2. Second solution:

   \[ x = \frac{-18 - 6}{6} = \frac{-24}{6} = -4 \]

Thus, the solutions to the equation are:

\[ x = -2 \quad \text{and} \quad x = -4 \]

Therefore, the correct response is:

x = −2, x = −4

Answer 6

Which of the following is a solution to the quadratic equation −x2+x=−10? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses

x=3.7
x equals 3.7

x=1.7
x equals 1.7

x=0.27
x equals 0.27

x=0.77

Answer 7

To find the solutions to the quadratic equation \(-x^2 + x = -10\), we first rearrange it into standard form:

\[ -x^2 + x + 10 = 0 \]

Next, we can multiply through by \(-1\) to make the leading coefficient positive:

\[ x^2 - x - 10 = 0 \]

Now, we can use the quadratic formula to find the solutions. The quadratic formula is given by:

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

For the equation \(x^2 - x - 10 = 0\), the coefficients are:

• \(a = 1\)
• \(b = -1\)
• \(c = -10\)

Now we calculate the discriminant \(b^2 - 4ac\):

\[ b^2 = (-1)^2 = 1 \] \[ 4ac = 4 \cdot 1 \cdot (-10) = -40 \] \[ \text{Discriminant} = 1 - (-40) = 1 + 40 = 41 \]

Now, we apply the quadratic formula:

\[ x = \frac{-(-1) \pm \sqrt{41}}{2 \cdot 1} \] \[ x = \frac{1 \pm \sqrt{41}}{2} \]

Calculating the two possible solutions:

1. First solution:

\[ x = \frac{1 + \sqrt{41}}{2} \]

Calculating \( \sqrt{41} \) (approximately \( 6.403 \)):

\[ x \approx \frac{1 + 6.403}{2} \approx \frac{7.403}{2} \approx 3.7015 \approx 3.70 \text{ (rounded to the nearest hundredth)} \]

2. Second solution:

\[ x = \frac{1 - \sqrt{41}}{2} \]

Calculating this yields:

\[ x \approx \frac{1 - 6.403}{2} \approx \frac{-5.403}{2} \approx -2.7015 \approx -2.70 \text{ (rounded to the nearest hundredth)} \]

The solutions, rounded to the nearest hundredth, are:

• \(x \approx 3.70\)
• \(x \approx -2.70\)

From the provided options, the closest and correct answer is:

x = 3.7

Answer 8

Which of the following coordinate pairs is a point on the parabola y=−4x2−53x−56?(1 point)
Responses

(−4,220)
left parenthesis negative 4 comma 220 right parenthesis

(−1,−7)
left parenthesis negative1 comma negative 7 right parenthesis

(−1,13)
left parenthesis neagtive 1 comma 13 right parenthesis

(−4,−56)
left parenthesis negative 4 comma negative 56 right parenthesis

Answer 9

To determine which coordinate pair is a point on the parabola given by the equation \( y = -4x^2 - 53x - 56 \), we will evaluate the equation for each \( x \)-coordinate in the given pairs and check if the corresponding \( y \)-coordinate matches.

1. For \( (-4, 220) \):

   \[ y = -4(-4)^2 - 53(-4) - 56 \] \[ y = -4(16) + 212 - 56 \] \[ y = -64 + 212 - 56 \] \[ y = 92 \quad (\text{not } 220) \]

2. For \( (-1, -7) \):

   \[ y = -4(-1)^2 - 53(-1) - 56 \] \[ y = -4(1) + 53 - 56 \] \[ y = -4 + 53 - 56 \] \[ y = -7 \quad (\text{matches}) \]

3. For \( (-1, 13) \):

   This has already been calculated for \( x = -1 \): \[ y = -7 \quad (\text{not } 13) \]

4. For \( (-4, -56) \):

   We know from the first calculation that: \[ y = 92 \quad (\text{not } -56) \]

In summary, the checks show:

• \( (-4, 220) \): \( y = 92 \) (not a point on the parabola)
• \( (-1, -7) \): \( y = -7 \) (a point on the parabola)
• \( (-1, 13) \): \( y = -7 \) (not a point on the parabola)
• \( (-4, -56) \): \( y = 92 \) (not a point on the parabola)

Thus, the correct response that is a point on the parabola is:

(−1, −7).