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Chapter 2: Problem 50

Solve each quadratic equation using the quadratic formula. Express solutions in standard form. $$3 x^{2}=4 x-6$$

Short Answer

Expert verified
The solutions of the quadratic equation \(3x^{2}=4x-6\) in standard form are \(x = \frac{2}{3} \pm \frac{i\sqrt{14}}{3}\)

Step by step solution

01

Re-arrange the equation into standard form

First, take all terms to one side of the equation to set it into standard form. The given equation is \(3x^{2}=4x-6\). Subtracting \(4x\) and adding \(6\) on both sides to make the equation as \(3x^{2}-4x+6 = 0\). Hence a = 3, b = -4 and c = 6.
02

Substitute values into the quadratic formula

In this step, we'll substitute the values of a, b and c into the quadratic formula. Doing that we get \(x = \frac{-(-4) \pm \sqrt{(-4)^{2}-4*3*6}}{2*3} = \frac{4 \pm \sqrt{16-72}}{6}.\)
03

Simplify the equation

Simplified further, the equation becomes \(x = \frac{4 \pm \sqrt{-56}}{6}\). Notice that under the square root sign we have a negative number. Since the square root of a negative number would lead to a complex number, our solutions will be in complex form.
04

Break down the square root of the negative number

The square root of \(-56\) can be broken down into \(\sqrt{-1*56}\) which is further simplified into \(\sqrt{-1}*\sqrt{56}\). Knowing that \(\sqrt{-1} = i\), (where i is an imaginary unit) we can represent the equation as \(x = \frac{4 \pm i\sqrt{56}}{6}\).
05

Simplify the equation further

Lastly, divide both the real and imaginary parts by 6 separately. Simplifying further we get the solutions as \(x = \frac{2}{3} \pm \frac{i\sqrt{14}}{3}\). This is the final answer in standard form.

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