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Solve 5x² - 11x - 12 = 0

The solutions of 5x² - 11x - 12 = 0 are x = 3 or x = -4/5.

x² +x += 0
5x2−11x−12=0⇒x=3, −455x^2 - 11x - 12 = 0 \Rightarrow x = 3,\ -\frac{4}{5}
Discriminant D = 361 · x ≈ 3, -0.8

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

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Graph of y = 5x² - 11x - 12-3-2-112345-2020406080x = 3x = -0.8vertexy = 5x² - 11x - 12
Graph of y = 5x² - 11x - 12. The curve crosses the x-axis at x = 3 and x = -0.8; the vertex is at (1.1, -18.05).

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 = -11, c = -12.

    5x2−11x−12=05x^2 - 11x - 12 = 0
  2. 2

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

    D=(−11)2−4⋅5⋅(−12)=121+240=361D = (-11)^2 - 4\cdot5\cdot(-12) = 121 + 240 = 361
  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+4)=0\left(x - 3\right)\left(5x + 4\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+4=0⇒x=−45x - 3 = 0 \Rightarrow x = 3\qquad 5x + 4 = 0 \Rightarrow x = -\frac{4}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±36110x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{361}}{10}
  7. 7

    Simplify to get the solutions:

    x=3orx=−45x = 3\quad\text{or}\quad x = -\frac{4}{5}
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = 5·(9) + (-11)·(3) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 11x - 12 = 0\ \Rightarrow\ x = 3,\ -\frac{4}{5} \]
\[ 5x^2 - 11x - 12 = 0 \]
\[ D = (-11)^2 - 4\cdot5\cdot(-12) = 121 + 240 = 361 \]
\[ \left(x - 3\right)\left(5x + 4\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad 5x + 4 = 0 \Rightarrow x = -\frac{4}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{361}}{10} \]
\[ x = 3\quad\text{or}\quad x = -\frac{4}{5} \]

The parabola y = 5x² - 11x - 12

xy
-1.926.95
-0.91.95
0.1-13.05
1.1-18.05
2.1-13.05
3.11.95
4.126.95

Questions

What are the solutions of 5x² - 11x - 12 = 0?

The solutions of 5x² - 11x - 12 = 0 are x = 3 or x = -4/5.

Does 5x² - 11x - 12 = 0 factor?

Yes: (x-3)(5x+4) = 0.

What is the vertex of y = 5x² - 11x - 12?

(11/10, -361/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 11/5, product = c/a = -12/5.

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