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Solve 5x² - 7x - 12 = 0

The solutions of 5x² - 7x - 12 = 0 are x = 12/5 or x = -1.

x² +x += 0
5x2−7x−12=0⇒x=125, −15x^2 - 7x - 12 = 0 \Rightarrow x = \frac{12}{5},\ -1
Discriminant D = 289 · x ≈ 2.4, -1

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

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Graph of y = 5x² - 7x - 12-3-2-11234-2020406080x = 2.4x = -1vertexy = 5x² - 7x - 12
Graph of y = 5x² - 7x - 12. The curve crosses the x-axis at x = 2.4 and x = -1; the vertex is at (0.7, -14.45).

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 = -7, c = -12.

    5x2−7x−12=05x^2 - 7x - 12 = 0
  2. 2

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

    D=(−7)2−4⋅5⋅(−12)=49+240=289D = (-7)^2 - 4\cdot5\cdot(-12) = 49 + 240 = 289
  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:

    (5x−12)(x+1)=0\left(5x - 12\right)\left(x + 1\right) = 0
  5. 5

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

    5x−12=0⇒x=125x+1=0⇒x=−15x - 12 = 0 \Rightarrow x = \frac{12}{5}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=7±28910x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{289}}{10}
  7. 7

    Simplify to get the solutions:

    x=125orx=−1x = \frac{12}{5}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 12/5 back in: a·x² + b·x + c = 5·(144/25) + (-7)·(12/5) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 7x - 12 = 0\ \Rightarrow\ x = \frac{12}{5},\ -1 \]
\[ 5x^2 - 7x - 12 = 0 \]
\[ D = (-7)^2 - 4\cdot5\cdot(-12) = 49 + 240 = 289 \]
\[ \left(5x - 12\right)\left(x + 1\right) = 0 \]
\[ 5x - 12 = 0 \Rightarrow x = \frac{12}{5}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{289}}{10} \]
\[ x = \frac{12}{5}\quad\text{or}\quad x = -1 \]

The parabola y = 5x² - 7x - 12

xy
-2.330.55
-1.35.55
-0.3-9.45
0.7-14.45
1.7-9.45
2.75.55
3.730.55

Questions

What are the solutions of 5x² - 7x - 12 = 0?

The solutions of 5x² - 7x - 12 = 0 are x = 12/5 or x = -1.

Does 5x² - 7x - 12 = 0 factor?

Yes: (5x-12)(x+1) = 0.

What is the vertex of y = 5x² - 7x - 12?

(7/10, -289/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 7/5, product = c/a = -12/5.

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