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Solve -3x² + 4x - 1 = 0

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

x² +x += 0
−3x2+4x−1=0⇒x=13, 1-3x^2 + 4x - 1 = 0 \Rightarrow x = \frac{1}{3},\ 1
Discriminant D = 4 · x ≈ 0.333333, 1

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

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Graph of y = -3x² + 4x - 1-2-1123-25-20-15-10-5x = 0.33333333x = 1vertexy = -3x² + 4x - 1
Graph of y = -3x² + 4x - 1. The curve crosses the x-axis at x = 0.33333333 and x = 1; the vertex is at (0.66666667, 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 = 4, c = -1.

    −3x2+4x−1=0-3x^2 + 4x - 1 = 0
  2. 2

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

    D=42−4⋅(−3)⋅(−1)=16−12=4D = 4^2 - 4\cdot(-3)\cdot(-1) = 16 - 12 = 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−1)(x−1)=0-\left(3x - 1\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:

    3x−1=0⇒x=13x−1=0⇒x=13x - 1 = 0 \Rightarrow x = \frac{1}{3}\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

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

    Check by putting x = 1/3 back in: a·x² + b·x + c = -3·(1/9) + 4·(1/3) + (-1) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 4x - 1 = 0\ \Rightarrow\ x = \frac{1}{3},\ 1 \]
\[ -3x^2 + 4x - 1 = 0 \]
\[ D = 4^2 - 4\cdot(-3)\cdot(-1) = 16 - 12 = 4 \]
\[ -\left(3x - 1\right)\left(x - 1\right) = 0 \]
\[ 3x - 1 = 0 \Rightarrow x = \frac{1}{3}\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{4}}{-6} \]
\[ x = \frac{1}{3}\quad\text{or}\quad x = 1 \]

The parabola y = -3x² + 4x - 1

xy
-2.3333-26.666067
-1.3333-11.666267
-0.3333-2.6664667
0.66670.33333333
1.6667-2.6668667
2.6667-11.667067
3.6667-26.667267

Questions

What are the solutions of -3x² + 4x - 1 = 0?

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

Does -3x² + 4x - 1 = 0 factor?

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

What is the vertex of y = -3x² + 4x - 1?

(2/3, 1/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 4/3, product = c/a = 1/3.

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