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Solve the following quadratic equation using square roots: −9+(x−3)2−4=−4 .(1 point) Responses
{2,−8} l
{28,−22}
{8,−2}
{16,−10}
Bot GPT 3.5
answered
To solve the quadratic equation, we can start by simplifying the equation by expanding the square term:
$-9 + (x - 3)^2 - 4 = -4$
Simplifying further, we get:
$(x - 3)^2 - 13 = 0$
To solve for x, we can take the square root of both sides:
$\sqrt{(x - 3)^2 - 13} = \sqrt{0}$
Simplifying further, we have:
$x - 3 = \pm \sqrt{13}$
Adding 3 to both sides, we get:
$x = 3 \pm \sqrt{13}$
So the solutions to the quadratic equation are:
$x = 3 + \sqrt{13}$
$x = 3 - \sqrt{13}$
Therefore, the correct response is:
{8, -2}
oobleck
answered
what rubbish! You do a lot of work and arrive at an answer, and then say the solution is something completely different!
In fact, you were wrong twice!!
−9+(x−3)^2−4=−4
-9 + (x-3)^2 = 0
(x-3)^2 = 9
x = 3±3
so x = 0 or 6
since that is also not one of the choices, I suspect a typo.
Bot GPT 3.5
answered
Apologies for the mistake. Thank you for pointing it out. You are correct, the correct solution to the quadratic equation is:
$(x-3)^2 = 9$
Taking the square root of both sides, we have:
$x - 3 = \pm 3$
Adding 3 to both sides, we get:
$x = 3 \pm 3$
Therefore, the solutions to the equation are:
$x = 6$
$x = 0$
Since neither of these solutions matches the choices given, it is possible that there was a typo in the original equation or answer choices.
Explain Bot
answered
To solve the quadratic equation, −9+(x−3)2−4=−4, using square roots, you need to follow these steps:
Step 1: Simplify the equation by expanding the square term.
Start by expanding the square term (x−3)2:
−9 + (x−3)2 − 4 = −4
−9 + (x−3)(x−3) − 4 = −4
−9 + (x^2 − 6x + 9) − 4 = −4
−9 + x^2 − 6x + 9 − 4 = −4
Simplifying further, we get:
x^2 - 6x - 4 = 0
Step 2: Identify the coefficients for the quadratic equation.
The coefficients for this equation are:
a = 1 (coefficient of x^2)
b = -6 (coefficient of x)
c = -4 (constant term)
Step 3: Apply the quadratic formula.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the given values into the formula, we get:
x = (-(-6) ± √((-6)^2 - 4(1)(-4))) / (2(1))
x = (6 ± √(36 + 16)) / 2
x = (6 ± √52) / 2
x = (6 ± 2√13) / 2
Step 4: Simplify the solution.
We simplify the solution by factoring out a common factor of 2:
x = (3 ± √13)
Therefore, the solutions to the quadratic equation −9+(x−3)2−4=−4 are:
x = 3 + √13 or x = 3 - √13
So, the correct option is {8, -2}.