WordGPT

Solve x² - 3x - 12 = 0

The solutions of x² - 3x - 12 = 0 are x = (3 + √57)/2 or x = (3 - √57)/2, that is x ≈ 5.2749172 and x ≈ -2.2749172.

x² +x += 0
x2−3x−12=0⇒x=3+572, 3−572x^2 - 3x - 12 = 0 \Rightarrow x = \frac{3 + \sqrt{57}}{2},\ \frac{3 - \sqrt{57}}{2}
Discriminant D = 57 · x ≈ 5.27492, -2.27492

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

Explain it step by step in chatMake a short video of it
Graph of y = x² - 3x - 12-4-2246-101020x = 5.2749172x = -2.2749172vertexy = x² - 3x - 12
Graph of y = x² - 3x - 12. The curve crosses the x-axis at x = 5.2749172 and x = -2.2749172; the vertex is at (1.5, -14.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = -3, c = -12.

    x2−3x−12=0x^2 - 3x - 12 = 0
  2. 2

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

    D=(−3)2−4⋅1⋅(−12)=9+48=57D = (-3)^2 - 4\cdot1\cdot(-12) = 9 + 48 = 57
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=3±572x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{57}}{2}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=3+572orx=3−572x = \frac{3 + \sqrt{57}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{57}}{2}
  6. 6

    As decimals: x ≈ 5.2749172 and x ≈ -2.2749172.

  7. 7

    Check numerically: for x ≈ 5.2749172 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 3x - 12 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{57}}{2},\ \frac{3 - \sqrt{57}}{2} \]
\[ x^2 - 3x - 12 = 0 \]
\[ D = (-3)^2 - 4\cdot1\cdot(-12) = 9 + 48 = 57 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{57}}{2} \]
\[ x = \frac{3 + \sqrt{57}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{57}}{2} \]

The parabola y = x² - 3x - 12

xy
-1.5-5.25
-0.5-10.25
0.5-13.25
1.5-14.25
2.5-13.25
3.5-10.25
4.5-5.25

Questions

What are the solutions of x² - 3x - 12 = 0?

The solutions of x² - 3x - 12 = 0 are x = (3 + √57)/2 or x = (3 - √57)/2, that is x ≈ 5.2749172 and x ≈ -2.2749172.

Does x² - 3x - 12 = 0 factor?

Not over the integers, because the discriminant 57 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 3x - 12?

(3/2, -57/4). 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 = -12.

Similar equations

    Solve x² - 3x - 12 = 0: x = (3 + √57)/2 or x = (3 - √57)/2 (steps) | WordGPT