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Solve -3x² - 8x + 3 = 0

The solutions of -3x² - 8x + 3 = 0 are x = -3 or x = 1/3.

x² +x += 0
−3x2−8x+3=0⇒x=−3, 13-3x^2 - 8x + 3 = 0 \Rightarrow x = -3,\ \frac{1}{3}
Discriminant D = 100 · x ≈ -3, 0.333333

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

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Graph of y = -3x² - 8x + 3-5-4-3-2-112-40-30-20-1010x = -3x = 0.33333333vertexy = -3x² - 8x + 3
Graph of y = -3x² - 8x + 3. The curve crosses the x-axis at x = -3 and x = 0.33333333; the vertex is at (-1.3333333, 8.3333333).

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 = -8, c = 3.

    −3x2−8x+3=0-3x^2 - 8x + 3 = 0
  2. 2

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

    D=(−8)2−4⋅(−3)⋅3=64+36=100D = (-8)^2 - 4\cdot(-3)\cdot3 = 64 + 36 = 100
  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)(3x−1)=0-\left(x + 3\right)\left(3x - 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=−33x−1=0⇒x=13x + 3 = 0 \Rightarrow x = -3\qquad 3x - 1 = 0 \Rightarrow x = \frac{1}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=8±100−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{100}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−3orx=13x = -3\quad\text{or}\quad x = \frac{1}{3}
  8. 8

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

LaTeX source of the formulas on this page
\[ -3x^2 - 8x + 3 = 0\ \Rightarrow\ x = -3,\ \frac{1}{3} \]
\[ -3x^2 - 8x + 3 = 0 \]
\[ D = (-8)^2 - 4\cdot(-3)\cdot3 = 64 + 36 = 100 \]
\[ -\left(x + 3\right)\left(3x - 1\right) = 0 \]
\[ x + 3 = 0 \Rightarrow x = -3\qquad 3x - 1 = 0 \Rightarrow x = \frac{1}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{100}}{-6} \]
\[ x = -3\quad\text{or}\quad x = \frac{1}{3} \]

The parabola y = -3x² - 8x + 3

xy
-4.3333-18.666067
-3.3333-3.6662667
-2.33335.3335333
-1.33338.3333333
-0.33335.3331333
0.6667-3.6670667
1.6667-18.667267

Questions

What are the solutions of -3x² - 8x + 3 = 0?

The solutions of -3x² - 8x + 3 = 0 are x = -3 or x = 1/3.

Does -3x² - 8x + 3 = 0 factor?

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

What is the vertex of y = -3x² - 8x + 3?

(-4/3, 25/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -8/3, product = c/a = -1.

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