Algebra

x^2 - 2x + 10 = 0

Step-by-step solution with explanation

Final Answer

and (two complex conjugate solutions, no real solutions)

Step-by-step solution

1

Identify the quadratic coefficients

We match to the standard form . Picking out , , and lets us plug into the quadratic formula.
2

Check the discriminant for solution type

The discriminant tells us how many real solutions exist. Because , there are no real solutions — only two complex (imaginary) solutions.
3

Apply the quadratic formula

We plug , , and the discriminant into the quadratic formula. The means we get two separate solutions.
4

Simplify the square root of negative number

We use the fact that (the imaginary unit). So .
5

Simplify both solutions completely

We divide both the real part and the imaginary part by 2. This gives us the two complex solutions in standard form.

Understanding this problem

Learning Insight

When the discriminant () is negative, the parabola never crosses the x-axis, so there are no real solutions. Complex solutions always come in conjugate pairs like and — the imaginary parts are opposites of each other.

Quick Tip

Memorize the discriminant test: positive → 2 real solutions, zero → 1 real solution, negative → 2 complex conjugate solutions. Check it BEFORE doing the full formula to know what to expect.

Common Mistake

Students often write instead of . You cannot take the square root of a negative number and get a real number — you must factor out .