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Solve -3x² + 10x - 8 = 0

The solutions of -3x² + 10x - 8 = 0 are x = 4/3 or x = 2.

x² +x += 0
−3x2+10x−8=0⇒x=43, 2-3x^2 + 10x - 8 = 0 \Rightarrow x = \frac{4}{3},\ 2
Discriminant D = 4 · x ≈ 1.33333, 2

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

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Graph of y = -3x² + 10x - 8-11234-25-20-15-10-5x = 1.3333333x = 2vertexy = -3x² + 10x - 8
Graph of y = -3x² + 10x - 8. The curve crosses the x-axis at x = 1.3333333 and x = 2; the vertex is at (1.6666667, 0.33333333).

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 = 10, c = -8.

    −3x2+10x−8=0-3x^2 + 10x - 8 = 0
  2. 2

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

    D=102−4⋅(−3)⋅(−8)=100−96=4D = 10^2 - 4\cdot(-3)\cdot(-8) = 100 - 96 = 4
  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:

    −(3x−4)(x−2)=0-\left(3x - 4\right)\left(x - 2\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    3x−4=0⇒x=43x−2=0⇒x=23x - 4 = 0 \Rightarrow x = \frac{4}{3}\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−10±4−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{4}}{-6}
  7. 7

    Simplify to get the solutions:

    x=43orx=2x = \frac{4}{3}\quad\text{or}\quad x = 2
  8. 8

    Check by putting x = 4/3 back in: a·x² + b·x + c = -3·(16/9) + 10·(4/3) + (-8) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 10x - 8 = 0\ \Rightarrow\ x = \frac{4}{3},\ 2 \]
\[ -3x^2 + 10x - 8 = 0 \]
\[ D = 10^2 - 4\cdot(-3)\cdot(-8) = 100 - 96 = 4 \]
\[ -\left(3x - 4\right)\left(x - 2\right) = 0 \]
\[ 3x - 4 = 0 \Rightarrow x = \frac{4}{3}\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{4}}{-6} \]
\[ x = \frac{4}{3}\quad\text{or}\quad x = 2 \]

The parabola y = -3x² + 10x - 8

xy
-1.3333-26.666067
-0.3333-11.666267
0.6667-2.6664667
1.66670.33333333
2.6667-2.6668667
3.6667-11.667067
4.6667-26.667267

Questions

What are the solutions of -3x² + 10x - 8 = 0?

The solutions of -3x² + 10x - 8 = 0 are x = 4/3 or x = 2.

Does -3x² + 10x - 8 = 0 factor?

Yes: -1(3x-4)(x-2) = 0.

What is the vertex of y = -3x² + 10x - 8?

(5/3, 1/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 10/3, product = c/a = 8/3.

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