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Solve -3x² + 8x - 5 = 0

The solutions of -3x² + 8x - 5 = 0 are x = 1 or x = 5/3.

x² +x += 0
−3x2+8x−5=0⇒x=1, 53-3x^2 + 8x - 5 = 0 \Rightarrow x = 1,\ \frac{5}{3}
Discriminant D = 4 · x ≈ 1, 1.66667

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

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Graph of y = -3x² + 8x - 5-11234-25-20-15-10-5x = 1x = 1.6666667vertexy = -3x² + 8x - 5
Graph of y = -3x² + 8x - 5. The curve crosses the x-axis at x = 1 and x = 1.6666667; the vertex is at (1.3333333, 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 = 8, c = -5.

    −3x2+8x−5=0-3x^2 + 8x - 5 = 0
  2. 2

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

    D=82−4⋅(−3)⋅(−5)=64−60=4D = 8^2 - 4\cdot(-3)\cdot(-5) = 64 - 60 = 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:

    −(x−1)(3x−5)=0-\left(x - 1\right)\left(3x - 5\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−1=0⇒x=13x−5=0⇒x=53x - 1 = 0 \Rightarrow x = 1\qquad 3x - 5 = 0 \Rightarrow x = \frac{5}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±4−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{4}}{-6}
  7. 7

    Simplify to get the solutions:

    x=1orx=53x = 1\quad\text{or}\quad x = \frac{5}{3}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = -3·(1) + 8·(1) + (-5) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 8x - 5 = 0\ \Rightarrow\ x = 1,\ \frac{5}{3} \]
\[ -3x^2 + 8x - 5 = 0 \]
\[ D = 8^2 - 4\cdot(-3)\cdot(-5) = 64 - 60 = 4 \]
\[ -\left(x - 1\right)\left(3x - 5\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 3x - 5 = 0 \Rightarrow x = \frac{5}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{4}}{-6} \]
\[ x = 1\quad\text{or}\quad x = \frac{5}{3} \]

The parabola y = -3x² + 8x - 5

xy
-1.6667-26.667267
-0.6667-11.667067
0.3333-2.6668667
1.33330.33333333
2.3333-2.6664667
3.3333-11.666267
4.3333-26.666067

Questions

What are the solutions of -3x² + 8x - 5 = 0?

The solutions of -3x² + 8x - 5 = 0 are x = 1 or x = 5/3.

Does -3x² + 8x - 5 = 0 factor?

Yes: -1(x-1)(3x-5) = 0.

What is the vertex of y = -3x² + 8x - 5?

(4/3, 1/3). 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 = 5/3.

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