QuestionRewrite $y=3 x^{2}-6 x+5$ as $a(x-h)^{2}+k$ with real constants $a, h, k$ .

Studdy Solution

STEP 1

Assumptions1. The given equation is $y=3x^-6x+5$ . We need to express this equation in the form $a(x-h)^+k$ , where $a$ , $h$ , and $k$ are real constants.

STEP 2

The equation $a(x-h)^2+k$ is the standard form of a parabola. To convert the given equation into this form, we need to complete the square.

STEP 3

First, we factor out the coefficient of $x^2$ from the first two terms of the given equation.
$y=3(x^2-2x)+5$

STEP 4

Next, we complete the square inside the parentheses by adding and subtracting the square of half the coefficient of $x$ inside the parentheses.
$y=3[(x^2-2x+1)-1]+$

STEP 5

implify the equation by combining like terms and simplifying the expression inside the parentheses.
$y=3[(x-1)^2-1]+5$

STEP 6

istribute the3 to both terms inside the parentheses.
$y=3(x-1)^2-3+5$

STEP 7

Combine the constants outside the parentheses.
$y=3(x-1)^2+2$

STEP 8

Now we have the equation in the form $a(x-h)^2+k$ , where $a=3$ , $h=1$ , and $k=2$ .
So, $y=3(x-1)^2+2$ is the equation in the form $a(x-h)^2+k$ .

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