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QuestionSolve x2+4x=−1x^{2}+4 x=-1. Hint: Add 4 to both sides to form a perfect square.

Studdy Solution

STEP 1

Assumptions1. The equation is x+4x=−1x^{}+4 x=-1 . We are solving for xx
3. We are allowed to perform the same operation on both sides of the equation

STEP 2

We can start by adding1 to both sides of the equation to move the -1 from the right side to the left side.
x2+4x+1=0x^{2}+4 x +1 =0

STEP 3

We can see that the left side of the equation is a perfect square trinomial. A perfect square trinomial is a trinomial that is the square of a binomial. In this case, the binomial is (x+2)(x+2), because (x+2)2=x2+x+(x+2)^2 = x^2 +x +.
x2+x+1=(x+2)2−3x^{2}+ x +1 = (x+2)^2 -3

STEP 4

Now, we can solve for xx by taking the square root of both sides of the equation. Remember, when we take the square root of both sides, we must consider both the positive and negative roots.
x+2=±−3x+2 = \pm \sqrt{-3}

STEP 5

Subtract2 from both sides to isolate xx.
x=−2±−3x = -2 \pm \sqrt{-3}

STEP 6

We can simplify −3\sqrt{-3} by recognizing that the square root of a negative number is an imaginary number. The square root of -1 is represented by the imaginary unit ii, so −3=3i\sqrt{-3} = \sqrt{3}i.
x=−2±3ix = -2 \pm \sqrt{3}iThe solutions are x=−2+3ix = -2 + \sqrt{3}i and x=−2−3ix = -2 - \sqrt{3}i.

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