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Solve -3x² + 7x - 4 = 0

The solutions of -3x² + 7x - 4 = 0 are x = 1 or x = 4/3.

x² +x += 0
−3x2+7x−4=0⇒x=1, 43-3x^2 + 7x - 4 = 0 \Rightarrow x = 1,\ \frac{4}{3}
Discriminant D = 1 · x ≈ 1, 1.33333

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

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Graph of y = -3x² + 7x - 4-1123-20-15-10-5x = 1x = 1.3333333vertexy = -3x² + 7x - 4
Graph of y = -3x² + 7x - 4. The curve crosses the x-axis at x = 1 and x = 1.3333333; the vertex is at (1.1666667, 0.083333333).

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 = 7, c = -4.

    −3x2+7x−4=0-3x^2 + 7x - 4 = 0
  2. 2

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

    D=72−4⋅(−3)⋅(−4)=49−48=1D = 7^2 - 4\cdot(-3)\cdot(-4) = 49 - 48 = 1
  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−4)=0-\left(x - 1\right)\left(3x - 4\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−4=0⇒x=43x - 1 = 0 \Rightarrow x = 1\qquad 3x - 4 = 0 \Rightarrow x = \frac{4}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−7±1−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{1}}{-6}
  7. 7

    Simplify to get the solutions:

    x=1orx=43x = 1\quad\text{or}\quad x = \frac{4}{3}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = -3·(1) + 7·(1) + (-4) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 7x - 4 = 0\ \Rightarrow\ x = 1,\ \frac{4}{3} \]
\[ -3x^2 + 7x - 4 = 0 \]
\[ D = 7^2 - 4\cdot(-3)\cdot(-4) = 49 - 48 = 1 \]
\[ -\left(x - 1\right)\left(3x - 4\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 3x - 4 = 0 \Rightarrow x = \frac{4}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{1}}{-6} \]
\[ x = 1\quad\text{or}\quad x = \frac{4}{3} \]

The parabola y = -3x² + 7x - 4

xy
-1.8333-26.916067
-0.8333-11.916267
0.1667-2.9164667
1.16670.08333333
2.1667-2.9168667
3.1667-11.917067
4.1667-26.917267

Questions

What are the solutions of -3x² + 7x - 4 = 0?

The solutions of -3x² + 7x - 4 = 0 are x = 1 or x = 4/3.

Does -3x² + 7x - 4 = 0 factor?

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

What is the vertex of y = -3x² + 7x - 4?

(7/6, 1/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 7/3, product = c/a = 4/3.

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