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Solve for [tex]x[/tex]:

[tex]\[3x = 6x - 2\][/tex]



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Points]
DETAILS
MY NOTES
OSELEMALG2 7.6.336.
Ive. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.
[tex]$
(p-7)(p+3)=-9
$[/tex]
[tex]$\square$[/tex]
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Response:

Solve for [tex]p[/tex]. Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.

[tex]\[(p - 7)(p + 3) = -9\][/tex]

Sagot :

GinnyAnswer
To solve the equation [tex]\((p - 7)(p + 3) = -9\)[/tex], follow these steps:

1. Expand the left-hand side:

First, let's expand the product [tex]\((p - 7)(p + 3)\)[/tex].
[tex]\[ (p - 7)(p + 3) = p(p + 3) - 7(p + 3) \][/tex]
Simplifying this, we get:
[tex]\[ = p^2 + 3p - 7p - 21 \][/tex]
Combining like terms:
[tex]\[ = p^2 - 4p - 21 \][/tex]

2. Set the equation equal to [tex]\(-9\)[/tex]:

Now, we set this expanded equation equal to [tex]\(-9\)[/tex]:
[tex]\[ p^2 - 4p - 21 = -9 \][/tex]

3. Move all terms to one side to set the equation to 0:

Add 9 to both sides to get a standard quadratic equation:
[tex]\[ p^2 - 4p - 21 + 9 = 0 \][/tex]
Simplifying further:
[tex]\[ p^2 - 4p - 12 = 0 \][/tex]

4. Factor the quadratic equation:

Next, we need to factor the quadratic equation [tex]\(p^2 - 4p - 12\)[/tex]. We look for two numbers that multiply to [tex]\(-12\)[/tex] and add to [tex]\(-4\)[/tex]. These numbers are [tex]\(-6\)[/tex] and [tex]\(2\)[/tex]:
[tex]\[ p^2 - 4p - 12 = (p - 6)(p + 2) = 0 \][/tex]

5. Solve for [tex]\(p\)[/tex]:

Set each factor to zero and solve for [tex]\(p\)[/tex]:
[tex]\[ p - 6 = 0 \quad \text{or} \quad p + 2 = 0 \][/tex]
Solving these:
[tex]\[ p = 6 \quad \text{or} \quad p = -2 \][/tex]

Therefore, the solutions to the equation [tex]\((p - 7)(p + 3) = -9\)[/tex] are [tex]\(p = 6\)[/tex] and [tex]\(p = -2\)[/tex].

Thus, the final answer is:
[tex]\[ \boxed{6, -2} \][/tex]
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