Select all of the following that are solutions to the equation x2+3x−10=0

Question

. There may be more than one solution.(2 points)
Responses

2
2

10
10

5
5

−5

Answer 1

To find the solutions to the equation \( x^2 + 3x - 10 = 0 \), we can use the quadratic formula:

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

In this case, the coefficients are:

• \( a = 1 \)
• \( b = 3 \)
• \( c = -10 \)

We can now substitute these values into the quadratic formula:

1. Calculate the discriminant: \[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot (-10) = 9 + 40 = 49 \]

2. Use the quadratic formula: \[ x = \frac{{-3 \pm \sqrt{49}}}{2 \cdot 1} = \frac{{-3 \pm 7}}{2} \]

Now, calculate the two potential solutions:

• For \( -3 + 7 \): \[ x_1 = \frac{{4}}{2} = 2 \]

• For \( -3 - 7 \): \[ x_2 = \frac{{-10}}{2} = -5 \]

Thus, the solutions to the equation \( x^2 + 3x - 10 = 0 \) are \( x = 2 \) and \( x = -5 \).

So, the correct responses are:

• 2
• -5

Answer 2

Solve the quadratic equation (x−4)2=25
. Enter both solutions for x
:(2 points)

Answer 3

To solve the equation \( (x - 4)^2 = 25 \), we can take the square root of both sides. However, we need to remember that taking the square root of a squared term can yield both a positive and a negative solution.

1. Start by taking the square root of both sides: \[ x - 4 = \pm 5 \]

2. This gives us two equations to solve for \( x \):

   • Case 1: \( x - 4 = 5 \)
   • Case 2: \( x - 4 = -5 \)

Case 1: \[ x - 4 = 5 \] \[ x = 5 + 4 = 9 \]

Case 2: \[ x - 4 = -5 \] \[ x = -5 + 4 = -1 \]

Thus, the two solutions for \( x \) are: \[ x = 9 \quad \text{and} \quad x = -1 \]

Answer 4

Use the Zero Product Property to find the possible values of x
for the equation (x−6)(x+4)=0
. Enter both solutions:(2 points)

Answer 5

To find the possible values of \( x \) for the equation \( (x - 6)(x + 4) = 0 \), we use the Zero Product Property, which states that if the product of two factors equals zero, then at least one of the factors must equal zero.

We can set each factor equal to zero:

1. First factor: \[ x - 6 = 0 \] Solving for \( x \) gives: \[ x = 6 \]

2. Second factor: \[ x + 4 = 0 \] Solving for \( x \) gives: \[ x = -4 \]

Thus, the two possible values of \( x \) are: \[ x = 6 \quad \text{and} \quad x = -4 \]

Answer 6

What is the greatest common factor of the expression 3x2−18x
?(1 point)
Responses

3x2
3 x squared

3
3

3x
3 x

x2
x squared
Question 2
What are the solutions of x
in the equation 3x2−18x=0
?(2 points)

Answer 7

Question 1: Greatest Common Factor (GCF)

To find the greatest common factor of the expression \( 3x^2 - 18x \), we can factor it.

1. Identify the coefficients and variables:

   • The coefficients are \( 3 \) and \( -18 \).
   • The variable parts are \( x^2 \) and \( x \).

2. The GCF of the coefficients \( 3 \) and \( -18 \) is \( 3 \).

3. The minimum power of \( x \) in \( x^2 \) and \( x \) is \( x \).

4. Therefore, the GCF of the expression \( 3x^2 - 18x \) is: \[ 3x \]

Responses for Question 1:

The answer is:

• 3x (option: 3 x)

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Question 2: Solutions for the equation \( 3x^2 - 18x = 0 \)

To solve the equation \( 3x^2 - 18x = 0 \), we can factor out the GCF, which we found is \( 3x \):

1. Factor the expression: \[ 3x(x - 6) = 0 \]

2. Now use the Zero Product Property and set each factor equal to zero:

   • Factor 1: \[ 3x = 0 \implies x = 0 \]
   • Factor 2: \[ x - 6 = 0 \implies x = 6 \]

The solutions for \( x \) in the equation \( 3x^2 - 18x = 0 \) are: \[ x = 0 \quad \text{and} \quad x = 6 \]

Responses for Question 2:

The solutions are:

• \( x = 0 \) and \( x = 6 \)

Answer 8

Fill in the blanks to write x2+14x+24=0
in its factored form.(2 points)
(x+

)(x+
)=0

Answer 9

To factor the equation \( x^2 + 14x + 24 = 0 \), we need to find two numbers that multiply to \( 24 \) (the constant term) and add up to \( 14 \) (the coefficient of \( x \)).

The pairs of factors of \( 24 \) are:

• \( 1 \times 24 \)
• \( 2 \times 12 \)
• \( 3 \times 8 \)
• \( 4 \times 6 \)

Among these pairs, the numbers \( 2 \) and \( 12 \) add up to \( 14 \).

Thus, we can write the factored form of the expression as:

\[ (x + 2)(x + 12) = 0 \]

Filling in the blanks:

So, the complete factored form is: \[ (x + 2)(x + 12) = 0 \]

So, you would fill the blanks as follows:

• First blank: 2
• Second blank: 12