Solve 5x² - 10x + 5 = 0
The solution of 5x² - 10x + 5 = 0 is x = 1 (a repeated root).
Change a, b or c (whole numbers, a not 0) and the exact solutions follow.
How to work it out
- 1
Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = -10, c = 5.
- 2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
- 3
D is zero, so there is exactly one solution (a repeated root).
- 4
Method 1, factoring. The roots are rational, so the left side factors:
- 5
A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:
- 6
Method 2, The quadratic formula works for every quadratic:
- 7
Simplify to get the solution:
- 8
Check by putting x = 1 back in: a·x² + b·x + c = 5·(1) + (-10)·(1) + 5 = 0 ✓
LaTeX source of the formulas on this page
\[ 5x^2 - 10x + 5 = 0\ \Rightarrow\ x = 1 \]
\[ 5x^2 - 10x + 5 = 0 \]
\[ D = (-10)^2 - 4\cdot5\cdot5 = 100 - 100 = 0 \]
\[ 5\left(x - 1\right)\left(x - 1\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{0}}{10} \]
\[ x = 1 \]The parabola y = 5x² - 10x + 5
| x | y |
|---|---|
| -2 | 45 |
| -1 | 20 |
| 0 | 5 |
| 1 | 0 |
| 2 | 5 |
| 3 | 20 |
| 4 | 45 |
Questions
What are the solutions of 5x² - 10x + 5 = 0?
The solution of 5x² - 10x + 5 = 0 is x = 1 (a repeated root).
Does 5x² - 10x + 5 = 0 factor?
Yes: 5(x-1)(x-1) = 0.
What is the vertex of y = 5x² - 10x + 5?
(1, 0). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 2, product = c/a = 1.