Solve 4x² - 12x + 9 = 0
The solution of 4x² - 12x + 9 = 0 is x = 3/2 (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 = 4, b = -12, c = 9.
- 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 = 3/2 back in: a·x² + b·x + c = 4·(9/4) + (-12)·(3/2) + 9 = 0 ✓
LaTeX source of the formulas on this page
\[ 4x^2 - 12x + 9 = 0\ \Rightarrow\ x = \frac{3}{2} \]
\[ 4x^2 - 12x + 9 = 0 \]
\[ D = (-12)^2 - 4\cdot4\cdot9 = 144 - 144 = 0 \]
\[ \left(2x - 3\right)\left(2x - 3\right) = 0 \]
\[ 2x - 3 = 0 \Rightarrow x = \frac{3}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{0}}{8} \]
\[ x = \frac{3}{2} \]The parabola y = 4x² - 12x + 9
| x | y |
|---|---|
| -1.5 | 36 |
| -0.5 | 16 |
| 0.5 | 4 |
| 1.5 | 0 |
| 2.5 | 4 |
| 3.5 | 16 |
| 4.5 | 36 |
Questions
What are the solutions of 4x² - 12x + 9 = 0?
The solution of 4x² - 12x + 9 = 0 is x = 3/2 (a repeated root).
Does 4x² - 12x + 9 = 0 factor?
Yes: (2x-3)(2x-3) = 0.
What is the vertex of y = 4x² - 12x + 9?
(3/2, 0). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 3, product = c/a = 9/4.