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Chapter 2: Problem 45

Solve each quadratic equation using the quadratic formula. Express solutions in standard form. $$x^{2}-6 x+10=0$$

Short Answer

Expert verified
The solutions of the given quadratic equation are \( x = 3 + i \) and \( x = 3 - i \).

Step by step solution

01

Identify the coefficients

In the given quadratic equation \(x^{2} - 6x + 10 = 0\), the coefficients are a=1, b=-6, and c=10.
02

Calculate the discriminant

The discriminant in the quadratic formula is \(b^{2} - 4ac = (-6)^{2} - 4(1)(10) = 36 - 40 = -4\). The discriminant is less than 0, which means the solutions will be complex.
03

Apply the Quadratic Formula

Solving for x using the quadratic formula gives: \(x = \frac{-(-6) \pm \sqrt{-4}}{2 * 1}\) = \(\frac{6 \pm \sqrt{-4}}{2}\) = \(3 \pm \frac{\sqrt{-4}}{2}\) = \(3 \pm \frac{\sqrt{4i^{2}}}{2}\) = \(3 \pm i\). Thus, the solutions are \(x = 3 + i\) and \(x = 3 - i\).

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