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Chapter 2: Problem 49

Solve each quadratic equation using the quadratic formula. Express solutions in standard form. $$3 x^{2}=8 x-7$$

Short Answer

Expert verified
The solutions to the equation \(3x^{2}=8x-7\) are \(x = 4/3 + i\sqrt{5}/3\) and \(x = 4/3 - i\sqrt{5}/3\)

Step by step solution

01

Identifying Values

First, arrange the given equation in the standard form of a quadratic equation, which is \(ax^{2}+bx+c=0\). Doing so gives \(3x^{2}-8x+7=0\). From this we can identify: \(a=3\), \(b=-8\), and \(c=7\).
02

Substitute into Quadratic Formula

Next, substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x=(-(-8)\pm \sqrt{(-8)^{2}-4*3*7})/(2*3)\).
03

Solve under the Square Root

First, resolve what's inside the square root in the formula. So, \( (-8^{2}) - (4*3*7) = 64-84 = -20 \), making the equation to now be \( x=(8\pm \sqrt{-20})/6 \)
04

Identify Complex Solution

Since the value under the square root is negative, we have a complex solution. Splitting this up gives \( x=(8\pm \sqrt{-1*20})/6 = 4/3 \pm (i*\sqrt{20})/6 \)
05

Simplify Complex Solution

Now, we simplify the complex solution, which gives \( x=4/3 \pm (\sqrt{4}i*\sqrt{5})/6 \) = \( 4/3 \pm (2i\sqrt{5})/6 \), or finally, \(x = 4/3 \pm i\sqrt{5}/3\)

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