Math / Algebra

QuestionSolve by completing the square. $h^{2}-6 h=41$
Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $h=$ $\square$ or $h=$ $\square$ Submits

Studdy Solution

STEP 1

What is this asking? We're asked to solve a quadratic equation by completing the square, meaning we need to manipulate the equation into a perfect square trinomial form and then solve for $h$ . Watch out! Remember that when we complete the square, we're adding a value to both sides of the equation to keep it balanced!
Also, don't forget to consider both positive and negative square roots when solving.

STEP 2

1. Set up the equation for completing the square.
2. Complete the square.
3. Isolate the squared term.
4. Take the square root of both sides.
5. Solve for $h$ .

STEP 3

Alright, let's get this party started!
We already have our equation set up nicely with the $h$ terms on one side and the constant on the other: $h^2 - 6h = 41$

STEP 4

Now, let's complete the square!
We take half of the coefficient of our $h$ term, which is $-6$ , so half of that is $-3$ .
Then, we square it: $(-3)^2 = 9$ .
This magical number, $9$ , is what we add to both sides of the equation: $h^2 - 6h + 9 = 41 + 9$

STEP 5

This gives us: $h^2 - 6h + 9 = 50$ Now, the left side is a perfect square trinomial!

STEP 6

We can rewrite that beautiful left side as a squared binomial: $(h - 3)^2 = 50$ Look at that – much cleaner!

STEP 7

Time to unleash the square root on both sides!
Remember, we need to consider both the positive and negative square roots: $\sqrt{(h - 3)^2} = \pm\sqrt{50}$ This simplifies to: $h - 3 = \pm\sqrt{50}$

STEP 8

Let's simplify that $\sqrt{50}$ .
Since $50 = 25 \cdot 2$ , we have $\sqrt{50} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$ .
So, we get: $h - 3 = \pm 5\sqrt{2}$

STEP 9

Almost there!
We just need to isolate $h$ by adding $3$ to both sides: $h = 3 \pm 5\sqrt{2}$

STEP 10

This gives us two possible solutions for $h$ : $h = 3 + 5\sqrt{2}$ and $h = 3 - 5\sqrt{2}$

STEP 11

$h = 3 + 5\sqrt{2}$ or $h = 3 - 5\sqrt{2}$

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QuestionSolve by completing the square. $h^{2}-6 h=41$
Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $h=$ $\square$ or $h=$ $\square$ Submits

Studdy Solution

STEP 1

STEP 2

1. Set up the equation for completing the square.
2. Complete the square.
3. Isolate the squared term.
4. Take the square root of both sides.
5. Solve for $h$ .

STEP 3

Alright, let's get this party started!
We already have our equation set up nicely with the $h$ terms on one side and the constant on the other: $h^2 - 6h = 41$

STEP 4

Now, let's complete the square!
We take half of the coefficient of our $h$ term, which is $-6$ , so half of that is $-3$ .
Then, we square it: $(-3)^2 = 9$ .
This magical number, $9$ , is what we add to both sides of the equation: $h^2 - 6h + 9 = 41 + 9$

STEP 5

This gives us: $h^2 - 6h + 9 = 50$ Now, the left side is a perfect square trinomial!

STEP 6

We can rewrite that beautiful left side as a squared binomial: $(h - 3)^2 = 50$ Look at that – much cleaner!

STEP 7

Time to unleash the square root on both sides!
Remember, we need to consider both the positive and negative square roots: $\sqrt{(h - 3)^2} = \pm\sqrt{50}$ This simplifies to: $h - 3 = \pm\sqrt{50}$

STEP 8

Let's simplify that $\sqrt{50}$ .
Since $50 = 25 \cdot 2$ , we have $\sqrt{50} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$ .
So, we get: $h - 3 = \pm 5\sqrt{2}$

STEP 9

Almost there!
We just need to isolate $h$ by adding $3$ to both sides: $h = 3 \pm 5\sqrt{2}$

STEP 10

This gives us two possible solutions for $h$ : $h = 3 + 5\sqrt{2}$ and $h = 3 - 5\sqrt{2}$

STEP 11

$h = 3 + 5\sqrt{2}$ or $h = 3 - 5\sqrt{2}$

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QuestionSolve by completing the square. h2−6h=41h^{2}-6 h=41h2−6h=41Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. h=h=h= □\square□ or h=h=h= □\square□ Submits

Studdy Solution

STEP 1

STEP 2

1. Set up the equation for completing the square.2. Complete the square.3. Isolate the squared term.4. Take the square root of both sides.5. Solve for hhh.

STEP 3

Alright, let's **get this party started**!We already have our equation set up nicely with the hhh terms on one side and the constant on the other: h2−6h=41h^2 - 6h = 41h2−6h=41

STEP 4

STEP 5

This gives us: h2−6h+9=50h^2 - 6h + 9 = 50h2−6h+9=50 Now, the left side is a **perfect square trinomial**!

STEP 6

We can rewrite that **beautiful** left side as a squared binomial: (h−3)2=50(h - 3)^2 = 50(h−3)2=50 Look at that – much cleaner!

STEP 7

Time to **unleash the square root** on both sides!Remember, we need to consider both the positive and negative square roots: (h−3)2=±50\sqrt{(h - 3)^2} = \pm\sqrt{50}(h−3)2​=±50​ This simplifies to: h−3=±50h - 3 = \pm\sqrt{50}h−3=±50​

STEP 8

Let's **simplify** that 50\sqrt{50}50​.Since 50=25⋅250 = 25 \cdot 250=25⋅2, we have 50=25⋅2=52\sqrt{50} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}50​=25​⋅2​=52​.So, we get: h−3=±52h - 3 = \pm 5\sqrt{2}h−3=±52​

STEP 9

Almost there!We just need to **isolate** hhh by adding 333 to both sides: h=3±52h = 3 \pm 5\sqrt{2}h=3±52​

STEP 10

This gives us two possible solutions for hhh: h=3+52h = 3 + 5\sqrt{2}h=3+52​ and h=3−52h = 3 - 5\sqrt{2}h=3−52​

STEP 11

h=3+52h = 3 + 5\sqrt{2}h=3+52​ or h=3−52h = 3 - 5\sqrt{2}h=3−52​

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionSolve by completing the square. $h^{2}-6 h=41$
Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $h=$ $\square$ or $h=$ $\square$ Submits

1. Set up the equation for completing the square.
2. Complete the square.
3. Isolate the squared term.
4. Take the square root of both sides.
5. Solve for $h$ .

Alright, let's get this party started!
We already have our equation set up nicely with the $h$ terms on one side and the constant on the other: $h^2 - 6h = 41$

This gives us: $h^2 - 6h + 9 = 50$ Now, the left side is a perfect square trinomial!

We can rewrite that beautiful left side as a squared binomial: $(h - 3)^2 = 50$ Look at that – much cleaner!

Time to unleash the square root on both sides!
Remember, we need to consider both the positive and negative square roots: $\sqrt{(h - 3)^2} = \pm\sqrt{50}$ This simplifies to: $h - 3 = \pm\sqrt{50}$

Let's simplify that $\sqrt{50}$ .
Since $50 = 25 \cdot 2$ , we have $\sqrt{50} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$ .
So, we get: $h - 3 = \pm 5\sqrt{2}$

Almost there!
We just need to isolate $h$ by adding $3$ to both sides: $h = 3 \pm 5\sqrt{2}$

This gives us two possible solutions for $h$ : $h = 3 + 5\sqrt{2}$ and $h = 3 - 5\sqrt{2}$

$h = 3 + 5\sqrt{2}$ or $h = 3 - 5\sqrt{2}$