Solve x² - 6x + 10 = 0

x² - 6x + 10 = 0 has no real solutions. The two complex solutions are x = 3 + i and x = 3 - i.

x² +x += 0

x^2 - 6x + 10 = 0 \Rightarrow x = 3 + i,\ 3 - i\ \text{(complex)}

Discriminant D = -4

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

x^2 - 6x + 10 = 0\ \Rightarrow\ x = 3 + i,\ 3 - i

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Graph of y = x² - 6x + 10. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (3, 1).

How to work it out

1
Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = -6, c = 10.
$x^2 - 6x + 10 = 0$
2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
$D = (-6)^2 - 4\cdot1\cdot10 = 36 - 40 = -4$
3
D is negative, so there are no real solutions; the two solutions are complex numbers.
4
The quadratic formula works for every quadratic:
$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{-4}}{2}$
5
Simplify (pull the square out of the root) to get the solutions:
$x = 3 + i\quad\text{or}\quad x = 3 - i$

LaTeX source of the formulas on this page

\[ x^2 - 6x + 10 = 0\ \Rightarrow\ x = 3 + i,\ 3 - i \]
\[ x^2 - 6x + 10 = 0 \]
\[ D = (-6)^2 - 4\cdot1\cdot10 = 36 - 40 = -4 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{-4}}{2} \]
\[ x = 3 + i\quad\text{or}\quad x = 3 - i \]

The parabola y = x² - 6x + 10

x	y
0	10
1	5
2	2
3	1
4	2
5	5
6	10

Questions

What are the solutions of x² - 6x + 10 = 0?

x² - 6x + 10 = 0 has no real solutions. The two complex solutions are x = 3 + i and x = 3 - i.

Does x² - 6x + 10 = 0 factor?

Not over the integers, because the discriminant -4 is negative. Use the quadratic formula.

What is the vertex of y = x² - 6x + 10?

(3, 1). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = 10.

Solve x² - 6x + 10 = 0

How to work it out

The parabola y = x² - 6x + 10

Questions

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