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Chapter 6: Problem 77

Solve each equation by the method of your choice. \((2 x-6)(x+2)=5(x-1)-12\)

Short Answer

Expert verified
The equation \(2x^2-7x+29=0\) doesn't have any real solutions because the discriminant, \(b^2 - 4ac\) is less than zero.

Step by step solution

01

Distribute and Simplify

Firstly, distribute on both sides of the equation to remove parentheses. This yields: \(2x^2+4x-6x-12=5x-5-12\). Combine like terms to simplify further: \(2x^2-2x-12=5x-17\).
02

Rearrange the Equation

Move all terms to one side of the equation to get the quadratic equation in standard form \(ax^2 + bx + c = 0\). Thus we get: \(2x^2-2x-5x+12+17=0\) which simplifies to \(2x^2-7x+29=0\).
03

Solve with Quadratic Formula

Apply the quadratic formula \(x = [-b 卤 sqrt(b^2 - 4ac)]/(2a)\). Substituting the values \(a = 2\), \(b = -7\), and \(c = 29\) into the formula, we get: \(x = [7 卤 sqrt((-7)^2 - 4*2*29)]/(2*2)\). After calculation, it emerges there are no real solutions because \(b^2 - 4ac\) is negative, indicating that the roots of the equation are imaginary.

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