Given the equation $$ x^{2}-12 x+15=0 $$, what is the value of $$ p $$ when the equation is written in the form $$ (x-p)^{2}=q $$ ? type your answer...

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UpStudy AI · Accepted Answer

We start with the given equation:
$$
x^2 - 12x + 15 = 0
$$

**Step 1: Complete the Square**

Focus on the terms involving $$ x $$:
$$
x^2 - 12x
$$
To complete the square, take half of the coefficient of $$ x $$, which is $$ -12 $$. Half of $$ -12 $$ is $$ -6 $$. Thus, we add and subtract $$(-6)^2 = 36$$.

$$
x^2 - 12x = \left(x^2 - 12x + 36\right) - 36 = (x-6)^2 - 36
$$

**Step 2: Rewrite the Equation**

Substitute the above expression back into the original equation:
$$
(x-6)^2 - 36 + 15 = 0
$$

Combine $$ -36 $$ and $$ 15 $$:
$$
(x-6)^2 - 21 = 0
$$

**Step 3: Express in the Form $$(x-p)^2 = q$$**

Add $$ 21 $$ to both sides:
$$
(x-6)^2 = 21
$$

The equation is now in the desired form $$(x-p)^2 = q$$, where:
$$
p = 6 \quad \text{and} \quad q = 21
$$

Thus, the value of $$ p $$ is:
$$
\boxed{6}
$$
$$ p = 6 $$

Given the equation , what is the value of when the equation is written in the form ?
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