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Solve the factored quadratic equation (19x−3)(9+x)=0.(1 point)

Responses

The solution set is x={−9,3}.
The solution set is x equals left brace negative 9 comma 3 right brace .

The solution set is x={19/3,9}.
The solution set is x equals left brace Start Fraction 19 over 3 End Fraction comma 9 right brace .

The solution set is x={−9,19/3}.
The solution set is x equals left brace negative 9 comma Start Fraction 19 over 3 End Fraction right brace .

The solution set is x={−9,3/19}

11 answers

To solve the factored quadratic equation \((19x−3)(9+x)=0\), we can set each factor equal to zero:

  1. \(19x - 3 = 0\)
  2. \(9 + x = 0\)

Now, let's solve each equation:

  1. For \(19x - 3 = 0\): \[ 19x = 3 \ x = \frac{3}{19} \]

  2. For \(9 + x = 0\): \[ x = -9 \]

Thus, the solutions to the equation are \(x = -9\) and \(x = \frac{3}{19}\).

Therefore, the solution set is: \[ x = {-9, \frac{3}{19}} \]

Among the given options, the closest one is: The solution set is \(x = {−9, \frac{3}{19}}\).

Solve the quadratic equation 84x^2+16x=0 by factoring out the GCF.(1 point)
Responses

The solutions are x=−4 and x=−4/21.
The solutions are x equals negative 4 and x equals negative Start Fraction 4 over 21 End Fraction .

The solutions are x=−4/21 and x=0.
The solutions are x equals negative Start Fraction 4 over 21 End Fraction and x equals 0 .

The solutions are x=0 and x=4.
The solutions are x equals 0 and x equals 4 .

The solutions are x=0 and x=4/21.

To solve the quadratic equation \(84x^2 + 16x = 0\) by factoring out the greatest common factor (GCF), we first identify the GCF of the terms:

The GCF of \(84x^2\) and \(16x\) is \(4x\).

Now, we can factor out \(4x\): \[ 4x(21x + 4) = 0 \]

Next, we set each factor equal to zero:

  1. \(4x = 0\)
  2. \(21x + 4 = 0\)

Solving the first equation: \[ 4x = 0 \implies x = 0 \]

Now solving the second equation: \[ 21x + 4 = 0 \implies 21x = -4 \implies x = -\frac{4}{21} \]

Thus, the solutions to the equation are: \[ x = 0 \quad \text{and} \quad x = -\frac{4}{21} \]

Therefore, the correct response is: The solutions are \(x = 0\) and \(x = -\frac{4}{21}\).

Which of the following quadratic equations is not solvable by grouping?(1 point)
Responses

x^2−12x+35=0
x squared minus 12 x plus 35 equals 0

2x^2−2x−10=0
2 x squared minus 2 x minus 10 equals 0

2x^2+14x+12=0
2 x squared plus 14 x plus 12 equals 0

x^2−2x+1=0

To determine which of the given quadratic equations is not solvable by grouping, we need to look for the following characteristics:

  1. Quadratic equations in standard form: \(ax^2 + bx + c = 0\)
  2. Can be factored into two binomials that can be grouped.

Let's analyze each option:

  1. \(x^2 - 12x + 35 = 0\)

    • Factors to \((x - 5)(x - 7) = 0\).
    • Solvable by grouping.

  2. \(2x^2 - 2x - 10 = 0\)

    • Factoring out the common factor of 2 gives: \(2(x^2 - x - 5) = 0\).
    • The quadratic \(x^2 - x - 5\) does not factor nicely, so it doesn't lend itself to grouping.
    • Not easily solvable by grouping.

  3. \(2x^2 + 14x + 12 = 0\)

    • Factoring out the common factor of 2 gives: \(2(x^2 + 7x + 6) = 0\).
    • The quadratic \(x^2 + 7x + 6\) factors to \((x + 1)(x + 6) = 0\).
    • Solvable by grouping.

  4. \(x^2 - 2x + 1 = 0\)

    • This can be factored as \((x - 1)(x - 1) = 0\).
    • Solvable by grouping.

Based on the analysis, the quadratic equation that is not solvable by grouping is:

\(2x^2 - 2x - 10 = 0\).

Jedida solved the quadratic equation x^2−96=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: x^2+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 1; she added 4x to both sides instead of subtracting.
Step 1; she added 4 x to both sides instead of subtracting.

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 3; she did not apply the Zero Product Property correctly

Let's analyze Jedida's work step by step to check for any mistakes:

Original equation: \[ x^2 - 96 = 4x \]

Step 1: Jedida rewrote the equation as: \[ x^2 + 4x - 96 = 0 \] This is incorrect because she should have subtracted \(4x\) from both sides, not added it. The correct form after moving \(4x\) to the left side would be: \[ x^2 - 4x - 96 = 0 \]

Since she made an error in rearranging the equation, this means her subsequent steps are also based on an incorrect equation.

Step 2: She factored the equation as: \[ (x + 12)(x - 8) = 0 \] This factoring cannot be correct since she started with an incorrect polynomial from Step 1. The polynomial \(x^2 + 4x - 96\) does not factor into \((x + 12)(x - 8)\). The correct factoring, if done on \(x^2 - 4x - 96\), would be different.

Step 3: She applied the Zero Product Property correctly in step 3, but since her factors were incorrect, the solutions are not valid.

Given all of this analysis, it is evident that the first mistake occurred in Step 1.

So the response is:

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

Solve the following quadratic equation using the perfect square trinomial pattern: x^2+x+0.25=0 .(1 point)
Responses

x=0.25
x equals 0.25

x=−0.25
x equals negative 0.25

x=−0.5
x equals negative 0.5

x=0.5
x equals 0.5

To solve the quadratic equation \(x^2 + x + 0.25 = 0\) using the perfect square trinomial pattern, we can recognize that the equation can be expressed as such.

The equation can be rewritten as: \[ x^2 + x + \left(\frac{1}{2}\right)^2 = 0 \]

Here, \(\left(\frac{1}{2}\right)^2 = 0.25\), so we can express it as: \[ \left(x + \frac{1}{2}\right)^2 = 0 \]

Now we take the square root of both sides: \[ x + \frac{1}{2} = 0 \]

To solve for \(x\), we subtract \(\frac{1}{2}\) from both sides: \[ x = -\frac{1}{2} \]

Thus, the solution to the equation is: \[ x = -0.5 \]

So the correct response is:

x = -0.5.

Ginger was asked to solve 25x^2−196=0 by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point)
Responses

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

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

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

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

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

The equation can be rewritten in the form of a difference of squares: \[ 25x^2 - 196 = 0 \] This is indeed a difference of squares since it can be expressed as: \[ (5x)^2 - (14)^2 = 0 \]

Using the difference of squares factoring pattern \(a^2 - b^2 = (a - b)(a + b)\), we can factor it: \[ (5x - 14)(5x + 14) = 0 \]

Next, we apply the zero product property by setting each factor equal to zero:

  1. \(5x - 14 = 0\)
  2. \(5x + 14 = 0\)

Now solve for \(x\) in both cases:

  1. For \(5x - 14 = 0\): \[ 5x = 14 \implies x = \frac{14}{5} \]

  2. For \(5x + 14 = 0\): \[ 5x = -14 \implies x = -\frac{14}{5} \]

Thus, the solutions to the equation are: \[ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} \]

So the correct response is: x = -14/5, x = 14/5.

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