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Solve 4x² - 8x - 12 = 0

The solutions of 4x² - 8x - 12 = 0 are x = 3 or x = -1.

x² +x += 0
4x2−8x−12=0⇒x=3, −14x^2 - 8x - 12 = 0 \Rightarrow x = 3,\ -1
Discriminant D = 256 · x ≈ 3, -1

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

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Graph of y = 4x² - 8x - 12-3-2-112345-20204060x = 3x = -1vertexy = 4x² - 8x - 12
Graph of y = 4x² - 8x - 12. The curve crosses the x-axis at x = 3 and x = -1; the vertex is at (1, -16).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = -8, c = -12.

    4x2−8x−12=04x^2 - 8x - 12 = 0
  2. 2

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

    D=(−8)2−4⋅4⋅(−12)=64+192=256D = (-8)^2 - 4\cdot4\cdot(-12) = 64 + 192 = 256
  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:

    4(x−3)(x+1)=04\left(x - 3\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:

    x−3=0⇒x=3x+1=0⇒x=−1x - 3 = 0 \Rightarrow x = 3\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=8±2568x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{256}}{8}
  7. 7

    Simplify to get the solutions:

    x=3orx=−1x = 3\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = 4·(9) + (-8)·(3) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 8x - 12 = 0\ \Rightarrow\ x = 3,\ -1 \]
\[ 4x^2 - 8x - 12 = 0 \]
\[ D = (-8)^2 - 4\cdot4\cdot(-12) = 64 + 192 = 256 \]
\[ 4\left(x - 3\right)\left(x + 1\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{256}}{8} \]
\[ x = 3\quad\text{or}\quad x = -1 \]

The parabola y = 4x² - 8x - 12

xy
-220
-10
0-12
1-16
2-12
30
420

Questions

What are the solutions of 4x² - 8x - 12 = 0?

The solutions of 4x² - 8x - 12 = 0 are x = 3 or x = -1.

Does 4x² - 8x - 12 = 0 factor?

Yes: 4(x-3)(x+1) = 0.

What is the vertex of y = 4x² - 8x - 12?

(1, -16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 2, product = c/a = -3.

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