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Solve 5x² - 12x - 9 = 0

The solutions of 5x² - 12x - 9 = 0 are x = 3 or x = -3/5.

x² +x += 0
5x2−12x−9=0⇒x=3, −355x^2 - 12x - 9 = 0 \Rightarrow x = 3,\ -\frac{3}{5}
Discriminant D = 324 · x ≈ 3, -0.6

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

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Graph of y = 5x² - 12x - 9-3-2-112345-2020406080x = 3x = -0.6vertexy = 5x² - 12x - 9
Graph of y = 5x² - 12x - 9. The curve crosses the x-axis at x = 3 and x = -0.6; the vertex is at (1.2, -16.2).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = -12, c = -9.

    5x2−12x−9=05x^2 - 12x - 9 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=(−12)2−4⋅5⋅(−9)=144+180=324D = (-12)^2 - 4\cdot5\cdot(-9) = 144 + 180 = 324
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (x−3)(5x+3)=0\left(x - 3\right)\left(5x + 3\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x−3=0⇒x=35x+3=0⇒x=−35x - 3 = 0 \Rightarrow x = 3\qquad 5x + 3 = 0 \Rightarrow x = -\frac{3}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=12±32410x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{324}}{10}
  7. 7

    Simplify to get the solutions:

    x=3orx=−35x = 3\quad\text{or}\quad x = -\frac{3}{5}
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = 5·(9) + (-12)·(3) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 12x - 9 = 0\ \Rightarrow\ x = 3,\ -\frac{3}{5} \]
\[ 5x^2 - 12x - 9 = 0 \]
\[ D = (-12)^2 - 4\cdot5\cdot(-9) = 144 + 180 = 324 \]
\[ \left(x - 3\right)\left(5x + 3\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad 5x + 3 = 0 \Rightarrow x = -\frac{3}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{324}}{10} \]
\[ x = 3\quad\text{or}\quad x = -\frac{3}{5} \]

The parabola y = 5x² - 12x - 9

xy
-1.828.8
-0.83.8
0.2-11.2
1.2-16.2
2.2-11.2
3.23.8
4.228.8

Questions

What are the solutions of 5x² - 12x - 9 = 0?

The solutions of 5x² - 12x - 9 = 0 are x = 3 or x = -3/5.

Does 5x² - 12x - 9 = 0 factor?

Yes: (x-3)(5x+3) = 0.

What is the vertex of y = 5x² - 12x - 9?

(6/5, -81/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 12/5, product = c/a = -9/5.

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