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Solve the following quadratic equation using square roots: −9+(x−3)^2/−4=−4

.(1 point)
Responses

{16, −10}

{16, −10}

{−2, 8}

{−2, 8-}{28, −22}.

3 answers

To solve the equation \( -9 + \frac{(x - 3)^2}{-4} = -4 \), let's begin by simplifying it step-by-step.

  1. First, isolate the term containing the square: \[ -9 + \frac{(x - 3)^2}{-4} = -4 \] Add 9 to both sides: \[ \frac{(x - 3)^2}{-4} = -4 + 9 \] Simplifying the right side gives: \[ \frac{(x - 3)^2}{-4} = 5 \]

  2. Multiply both sides by -4 to eliminate the fraction: \[ (x - 3)^2 = -20 \]

  3. Notice that we have a situation where a square equals a negative number. However, the square of a real number cannot be negative. Thus, we analyze the outcome: \[ (x - 3)^2 = -20 \] This means there are no real solutions, but we can find complex solutions.

  4. To solve for \( x \) in terms of complex numbers, take the square root of both sides: \[ x - 3 = \pm \sqrt{-20} \] We can simplify \( \sqrt{-20} \) as: \[ \sqrt{-20} = \sqrt{20}i = 2\sqrt{5}i \]

  5. Now, solve for \( x \): \[ x - 3 = \pm 2\sqrt{5}i \] Thus, \[ x = 3 \pm 2\sqrt{5}i \]

Since this gives complex solutions, we do not have real solutions for \( x \).

The answer choices you provided appear to be for real solutions only, which would not apply here. However, the complex solutions we found do not correspond to the listed options.

Thus, there are no real solutions to the original equation; the solutions involve complex numbers.

If you need further clarification or help with complex numbers, feel free to ask!

Based on the Zero Product Property, if either of the two factors in a quadratic equation equals zero, the entire equation equals zero. Find the values of x
𝑥
that make the following quadratic equation equal 0: (x+4)(x−2)=0
(
𝑥
+
4
)
(
𝑥
−
2
)
=
0
(1 point)
Responses

x=−8
𝑥
=
−
8
and x=0
𝑥
=
0
x is equal to negative 8 and x is equal to 0

x=−4
𝑥
=
−
4
and x=2
𝑥
=
2
x is equal to negative 4 and x is equal to 2

x=4
𝑥
=
4
and x=−2
𝑥
=
−
2
x is equal to 4 and x is equal to negative 2

x=2

To solve the quadratic equation \((x + 4)(x - 2) = 0\) using the Zero Product Property, we set each factor equal to zero:

  1. Set the first factor to zero: \[ x + 4 = 0 \] Solving for \(x\): \[ x = -4 \]

  2. Set the second factor to zero: \[ x - 2 = 0 \] Solving for \(x\): \[ x = 2 \]

Thus, the values of \(x\) that make the equation equal zero are: \[ x = -4 \quad \text{and} \quad x = 2 \]

So, the correct response is:

x = -4 and x = 2 (x is equal to negative 4 and x is equal to 2).