Math  /  Algebra

QuestionSolve the quadratic equation by completing the square. x2+6x+4=0x^{2}+6 x+4=0
First, choose the appropriate form and fill in the blanks with the correct numbers. Then, solve the equation. Simplify your answer as much as possible. If there is more than one solution, separate them with commas.
Form: (x+)2=(x+\square)^{2}= \square (x)2=(x-\square)^{2}= \square Solution: x=x= \square

Studdy Solution

STEP 1

What is this asking? We need to solve x2+6x+4=0x^{2} + 6x + 4 = 0 by completing the square, which means we want to rewrite it in a neat form like (x+)2=(x + \square)^{2} = \square and then find the value(s) of xx. Watch out! Remember, when we take the square root of both sides of an equation, we need to consider both the positive and negative square roots!

STEP 2

1. Rewrite the equation.
2. Complete the square.
3. Isolate the squared term.
4. Take the square root of both sides.
5. Solve for xx.

STEP 3

Let's **rewrite** our equation x2+6x+4=0x^{2} + 6x + 4 = 0 by moving the constant term to the right-hand side.
We **subtract** 4 from both sides to get: x2+6x=4x^{2} + 6x = -4

STEP 4

Now, let's **complete the square** on the left side.
Take the **coefficient** of our xx term, which is **6**, divide it by **2** to get **3**, and then square the result to get **9**.
We'll **add** this value to both sides of our equation: x2+6x+9=4+9x^{2} + 6x + 9 = -4 + 9

STEP 5

The left side is now a **perfect square trinomial**, which we can **factor**: (x+3)2=5(x+3)^{2} = 5

STEP 6

Our squared term is already isolated, so we're good to go! (x+3)2=5(x+3)^{2} = 5

STEP 7

Let's **take the square root** of both sides of the equation.
Remember to consider both the positive and negative square root: (x+3)2=±5\sqrt{(x+3)^{2}} = \pm\sqrt{5} x+3=±5x+3 = \pm\sqrt{5}

STEP 8

Finally, **isolate** xx by **subtracting** 3 from both sides of the equation: x=3±5x = -3 \pm\sqrt{5}

STEP 9

Form: (x+3)2=5(x+3)^{2}=5 Solution: x=3+5,35x = -3 + \sqrt{5}, -3 - \sqrt{5}

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