Solved on Dec 06, 2023

Rewrite each equation in the form $y=a(x-h)^2+k$ by completing the square: a) $y=x^2+6x-1$ , b) $y=x^2+10x+20$ .

STEP 1

Assumptions
1. We need to rewrite the given quadratic equations in the form $y = a(x-h)^2 + k$ .
2. This process involves completing the square for each equation.
3. Completing the square is a method used to create a perfect square trinomial from a quadratic equation.

STEP 2

Starting with equation (a) $y = x^2 + 6x - 1$ , we will complete the square.

STEP 3

Identify the coefficient of $x$ which is 6 in this case.

STEP 4

Divide the coefficient of $x$ by 2 and square the result to find the number that will complete the square.
$\left(\frac{6}{2}\right)^2 = 3^2 = 9$

STEP 5

Add and subtract this number inside the equation to maintain equality.
$y = x^2 + 6x + 9 - 9 - 1$

STEP 6

Now, group the perfect square trinomial and the constants together.
$y = (x^2 + 6x + 9) - 9 - 1$

STEP 7

Factor the perfect square trinomial.
$y = (x + 3)^2 - 9 - 1$

STEP 8

Combine the constants.
$y = (x + 3)^2 - 10$

STEP 9

Now we have equation (a) in the form $y = a(x-h)^2 + k$ :
$y = (x + 3)^2 - 10$

STEP 10

Next, we will complete the square for equation (b) $y = x^2 + 10x + 20$ .

STEP 11

Identify the coefficient of $x$ which is 10 in this case.

STEP 12

Divide the coefficient of $x$ by 2 and square the result to find the number that will complete the square.
$\left(\frac{10}{2}\right)^2 = 5^2 = 25$

STEP 13

Add and subtract this number inside the equation to maintain equality.
$y = x^2 + 10x + 25 - 25 + 20$

STEP 14

Group the perfect square trinomial and the constants together.
$y = (x^2 + 10x + 25) - 25 + 20$

STEP 15

Factor the perfect square trinomial.
$y = (x + 5)^2 - 25 + 20$

STEP 16

Combine the constants.
$y = (x + 5)^2 - 5$

STEP 17

Now we have equation (b) in the form $y = a(x-h)^2 + k$ :
$y = (x + 5)^2 - 5$
The solutions for the given quadratic equations in the form $y = a(x-h)^2 + k$ are:
a) $y = (x + 3)^2 - 10$
b) $y = (x + 5)^2 - 5$

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Rewrite each equation in the form $y=a(x-h)^2+k$ by completing the square: a) $y=x^2+6x-1$ , b) $y=x^2+10x+20$ .

STEP 1

Assumptions
1. We need to rewrite the given quadratic equations in the form $y = a(x-h)^2 + k$ .
2. This process involves completing the square for each equation.
3. Completing the square is a method used to create a perfect square trinomial from a quadratic equation.

STEP 2

Starting with equation (a) $y = x^2 + 6x - 1$ , we will complete the square.

STEP 3

Identify the coefficient of $x$ which is 6 in this case.

STEP 4

Divide the coefficient of $x$ by 2 and square the result to find the number that will complete the square.
$\left(\frac{6}{2}\right)^2 = 3^2 = 9$

STEP 5

Add and subtract this number inside the equation to maintain equality.
$y = x^2 + 6x + 9 - 9 - 1$

STEP 6

Now, group the perfect square trinomial and the constants together.
$y = (x^2 + 6x + 9) - 9 - 1$

STEP 7

Factor the perfect square trinomial.
$y = (x + 3)^2 - 9 - 1$

STEP 8

Combine the constants.
$y = (x + 3)^2 - 10$

STEP 9

Now we have equation (a) in the form $y = a(x-h)^2 + k$ :
$y = (x + 3)^2 - 10$

STEP 10

Next, we will complete the square for equation (b) $y = x^2 + 10x + 20$ .

STEP 11

Identify the coefficient of $x$ which is 10 in this case.

STEP 12

Divide the coefficient of $x$ by 2 and square the result to find the number that will complete the square.
$\left(\frac{10}{2}\right)^2 = 5^2 = 25$

STEP 13

Add and subtract this number inside the equation to maintain equality.
$y = x^2 + 10x + 25 - 25 + 20$

STEP 14

Group the perfect square trinomial and the constants together.
$y = (x^2 + 10x + 25) - 25 + 20$

STEP 15

Factor the perfect square trinomial.
$y = (x + 5)^2 - 25 + 20$

STEP 16

Combine the constants.
$y = (x + 5)^2 - 5$

STEP 17

Now we have equation (b) in the form $y = a(x-h)^2 + k$ :
$y = (x + 5)^2 - 5$
The solutions for the given quadratic equations in the form $y = a(x-h)^2 + k$ are:
a) $y = (x + 3)^2 - 10$
b) $y = (x + 5)^2 - 5$

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Rewrite each equation in the form y=a(x−h)2+ky=a(x-h)^2+ky=a(x−h)2+k by completing the square: a) y=x2+6x−1y=x^2+6x-1y=x2+6x−1, b) y=x2+10x+20y=x^2+10x+20y=x2+10x+20.

STEP 1

Assumptions1. We need to rewrite the given quadratic equations in the form y=a(x−h)2+k y = a(x-h)^2 + k y=a(x−h)2+k.2. This process involves completing the square for each equation.3. Completing the square is a method used to create a perfect square trinomial from a quadratic equation.

STEP 2

Starting with equation (a) y=x2+6x−1 y = x^2 + 6x - 1 y=x2+6x−1, we will complete the square.

STEP 3

Identify the coefficient of x x x which is 6 in this case.

STEP 4

Divide the coefficient of x x x by 2 and square the result to find the number that will complete the square.(62)2=32=9\left(\frac{6}{2}\right)^2 = 3^2 = 9(26​)2=32=9

STEP 5

Add and subtract this number inside the equation to maintain equality.y=x2+6x+9−9−1y = x^2 + 6x + 9 - 9 - 1y=x2+6x+9−9−1

STEP 6

Now, group the perfect square trinomial and the constants together.y=(x2+6x+9)−9−1y = (x^2 + 6x + 9) - 9 - 1y=(x2+6x+9)−9−1

STEP 7

Factor the perfect square trinomial.y=(x+3)2−9−1y = (x + 3)^2 - 9 - 1y=(x+3)2−9−1

STEP 8

Combine the constants.y=(x+3)2−10y = (x + 3)^2 - 10y=(x+3)2−10

STEP 9

Now we have equation (a) in the form y=a(x−h)2+k y = a(x-h)^2 + k y=a(x−h)2+k:y=(x+3)2−10y = (x + 3)^2 - 10y=(x+3)2−10

STEP 10

Next, we will complete the square for equation (b) y=x2+10x+20 y = x^2 + 10x + 20 y=x2+10x+20.

STEP 11

Identify the coefficient of x x x which is 10 in this case.

STEP 12

Divide the coefficient of x x x by 2 and square the result to find the number that will complete the square.(102)2=52=25\left(\frac{10}{2}\right)^2 = 5^2 = 25(210​)2=52=25

STEP 13

Add and subtract this number inside the equation to maintain equality.y=x2+10x+25−25+20y = x^2 + 10x + 25 - 25 + 20y=x2+10x+25−25+20

STEP 14

Group the perfect square trinomial and the constants together.y=(x2+10x+25)−25+20y = (x^2 + 10x + 25) - 25 + 20y=(x2+10x+25)−25+20

STEP 15

Factor the perfect square trinomial.y=(x+5)2−25+20y = (x + 5)^2 - 25 + 20y=(x+5)2−25+20

STEP 16

Combine the constants.y=(x+5)2−5y = (x + 5)^2 - 5y=(x+5)2−5

STEP 17

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Rewrite each equation in the form $y=a(x-h)^2+k$ by completing the square: a) $y=x^2+6x-1$ , b) $y=x^2+10x+20$ .

Assumptions
1. We need to rewrite the given quadratic equations in the form $y = a(x-h)^2 + k$ .
2. This process involves completing the square for each equation.
3. Completing the square is a method used to create a perfect square trinomial from a quadratic equation.

Starting with equation (a) $y = x^2 + 6x - 1$ , we will complete the square.

Identify the coefficient of $x$ which is 6 in this case.

Divide the coefficient of $x$ by 2 and square the result to find the number that will complete the square.
$\left(\frac{6}{2}\right)^2 = 3^2 = 9$

Add and subtract this number inside the equation to maintain equality.
$y = x^2 + 6x + 9 - 9 - 1$

Now, group the perfect square trinomial and the constants together.
$y = (x^2 + 6x + 9) - 9 - 1$

Factor the perfect square trinomial.
$y = (x + 3)^2 - 9 - 1$

Combine the constants.
$y = (x + 3)^2 - 10$

Now we have equation (a) in the form $y = a(x-h)^2 + k$ :
$y = (x + 3)^2 - 10$

Next, we will complete the square for equation (b) $y = x^2 + 10x + 20$ .

Identify the coefficient of $x$ which is 10 in this case.

Divide the coefficient of $x$ by 2 and square the result to find the number that will complete the square.
$\left(\frac{10}{2}\right)^2 = 5^2 = 25$

Add and subtract this number inside the equation to maintain equality.
$y = x^2 + 10x + 25 - 25 + 20$

Group the perfect square trinomial and the constants together.
$y = (x^2 + 10x + 25) - 25 + 20$

Factor the perfect square trinomial.
$y = (x + 5)^2 - 25 + 20$

Combine the constants.
$y = (x + 5)^2 - 5$