Math

QuestionSolve for xx using the quadratic formula for the equation 4x2+12x+12=04x^{2} + 12x + 12 = 0.

Studdy Solution

STEP 1

Assumptions1. The given equation is 4x+12x=124x^ +12x = -12 . We are to solve for xx using the quadratic formula3. The quadratic formula is x=b±b4acax = \frac{-b \pm \sqrt{b^ -4ac}}{a} for any quadratic equation of the form ax+bx+c=0ax^ + bx + c =0

STEP 2

First, we need to rewrite the equation in the standard form ax2+bx+c=0ax^2 + bx + c =0. This can be done by adding12 to both sides of the equation.
4x2+12x+12=04x^2 +12x +12 =0

STEP 3

Now, we can identify the coefficients aa, bb, and cc in the equation.a=,b=12,c=12a =, b =12, c =12

STEP 4

Next, we substitute these coefficients into the quadratic formula.
x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}x=12±122441224x = \frac{-12 \pm \sqrt{12^2 -4*4*12}}{2*4}

STEP 5

implify the expression under the square root.
x=12±1441928x = \frac{-12 \pm \sqrt{144 -192}}{8}

STEP 6

Calculate the value under the square root.
x=12±488x = \frac{-12 \pm \sqrt{-48}}{8}

STEP 7

Since the value under the square root is negative, we have a complex solution. We can write the square root of -48 as i48i\sqrt{48}, where ii is the imaginary unit.
x=12±i48x = \frac{-12 \pm i\sqrt{48}}{}

STEP 8

implify the expression further by factoring out4 from under the square root.
x=12±4i38x = \frac{-12 \pm4i\sqrt{3}}{8}

STEP 9

Divide each term in the numerator by4.
x=3±i32x = \frac{-3 \pm i\sqrt{3}}{2}So the solutions to the equation are x=32+32ix = -\frac{3}{2} + \frac{\sqrt{3}}{2}i and x=3232ix = -\frac{3}{2} - \frac{\sqrt{3}}{2}i.

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