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Solve -3x² + 8x - 3 = 0

The solutions of -3x² + 8x - 3 = 0 are x = (4 + √7)/3 or x = (4 - √7)/3, that is x ≈ 0.45141623 and x ≈ 2.2152504.

x² +x += 0
−3x2+8x−3=0⇒x=4+73, 4−73-3x^2 + 8x - 3 = 0 \Rightarrow x = \frac{4 + \sqrt{7}}{3},\ \frac{4 - \sqrt{7}}{3}
Discriminant D = 28 · x ≈ 0.451416, 2.21525

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

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Graph of y = -3x² + 8x - 3-2-11234-30-25-20-15-10-55x = 0.45141623x = 2.2152504vertexy = -3x² + 8x - 3
Graph of y = -3x² + 8x - 3. The curve crosses the x-axis at x = 0.45141623 and x = 2.2152504; the vertex is at (1.3333333, 2.3333333).

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=82−4⋅(−3)⋅(−3)=64−36=28D = 8^2 - 4\cdot(-3)\cdot(-3) = 64 - 36 = 28
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−8±28−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{28}}{-6}
  5. 5

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

    x=4+73orx=4−73x = \frac{4 + \sqrt{7}}{3}\quad\text{or}\quad x = \frac{4 - \sqrt{7}}{3}
  6. 6

    As decimals: x ≈ 0.45141623 and x ≈ 2.2152504.

  7. 7

    Check numerically: for x ≈ 0.45141623 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 8x - 3 = 0\ \Rightarrow\ x = \frac{4 + \sqrt{7}}{3},\ \frac{4 - \sqrt{7}}{3} \]
\[ -3x^2 + 8x - 3 = 0 \]
\[ D = 8^2 - 4\cdot(-3)\cdot(-3) = 64 - 36 = 28 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{28}}{-6} \]
\[ x = \frac{4 + \sqrt{7}}{3}\quad\text{or}\quad x = \frac{4 - \sqrt{7}}{3} \]

The parabola y = -3x² + 8x - 3

xy
-1.6667-24.667267
-0.6667-9.6670667
0.3333-0.66686667
1.33332.3333333
2.3333-0.66646667
3.3333-9.6662667
4.3333-24.666067

Questions

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

The solutions of -3x² + 8x - 3 = 0 are x = (4 + √7)/3 or x = (4 - √7)/3, that is x ≈ 0.45141623 and x ≈ 2.2152504.

Does -3x² + 8x - 3 = 0 factor?

Not over the integers, because the discriminant 28 is not a perfect square. Use the quadratic formula.

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

(4/3, 7/3). 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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