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Chapter 2: Problem 47

Solve each quadratic equation using the quadratic formula. Express solutions in standard form. $$4 x^{2}+8 x+13=0$$

Short Answer

Expert verified
The solutions for the equation \(4x^2 + 8x + 13 = 0\) using the quadratic formula are \(-1 + 1.5i\) and \(-1 - 1.5i\).

Step by step solution

01

Identifying Coefficients

In the equation \(4x^2 + 8x + 13 = 0\), identify the coefficients a, b, and c, which are representative of the standard quadratic formula \(ax^2 + bx + c = 0\). For this equation, a = 4, b = 8, and c = 13.
02

Substitute values into quadratic formula

Substitute a, b, and c into the quadratic formula, which is \(-b \pm \sqrt {b^2 - 4ac} / 2a\). For this equation, the quadratic formula becomes: \(-8 \pm \sqrt {(8^2) - 4*4*13} / 2*4\)
03

Simplify the Expression

Simplify the expression under the root sign and the fractions. Consequently, it can be expressed as: \(-8 \pm \sqrt {64 - 208} / 8\). Further simplification results in: \(-8 \pm \sqrt {-144} / 8\). The root of a negative number produces an imaginary term. The sqrt(-144) can be written as 12i.
04

Write Solution in Standard Form

Now replace the \(\sqrt{-144}\) with 12i in the earlier equation, which results as \(-8 \pm 12i / 8\). Further simplifying this equation, one obtains \(-1 \pm 1.5i\) which is the solution in standard form.

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