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Solve -3x² - 9x + 12 = 0

The solutions of -3x² - 9x + 12 = 0 are x = -4 or x = 1.

x² +x += 0
−3x2−9x+12=0⇒x=−4, 1-3x^2 - 9x + 12 = 0 \Rightarrow x = -4,\ 1
Discriminant D = 225 · x ≈ -4, 1

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

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Graph of y = -3x² - 9x + 12-6-5-4-3-2-1123-60-40-2020x = -4x = 1vertexy = -3x² - 9x + 12
Graph of y = -3x² - 9x + 12. The curve crosses the x-axis at x = -4 and x = 1; the vertex is at (-1.5, 18.75).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -3, b = -9, c = 12.

    −3x2−9x+12=0-3x^2 - 9x + 12 = 0
  2. 2

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

    D=(−9)2−4⋅(−3)⋅12=81+144=225D = (-9)^2 - 4\cdot(-3)\cdot12 = 81 + 144 = 225
  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:

    −3(x+4)(x−1)=0-3\left(x + 4\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+4=0⇒x=−4x−1=0⇒x=1x + 4 = 0 \Rightarrow x = -4\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=9±225−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{225}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−4orx=1x = -4\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = -4 back in: a·x² + b·x + c = -3·(16) + (-9)·(-4) + 12 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 9x + 12 = 0\ \Rightarrow\ x = -4,\ 1 \]
\[ -3x^2 - 9x + 12 = 0 \]
\[ D = (-9)^2 - 4\cdot(-3)\cdot12 = 81 + 144 = 225 \]
\[ -3\left(x + 4\right)\left(x - 1\right) = 0 \]
\[ x + 4 = 0 \Rightarrow x = -4\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{225}}{-6} \]
\[ x = -4\quad\text{or}\quad x = 1 \]

The parabola y = -3x² - 9x + 12

xy
-4.5-8.25
-3.56.75
-2.515.75
-1.518.75
-0.515.75
0.56.75
1.5-8.25

Questions

What are the solutions of -3x² - 9x + 12 = 0?

The solutions of -3x² - 9x + 12 = 0 are x = -4 or x = 1.

Does -3x² - 9x + 12 = 0 factor?

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

What is the vertex of y = -3x² - 9x + 12?

(-3/2, 75/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = -4.

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