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Solve 3x² - 4x - 7 = 0

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

x² +x += 0
3x2−4x−7=0⇒x=73, −13x^2 - 4x - 7 = 0 \Rightarrow x = \frac{7}{3},\ -1
Discriminant D = 100 · x ≈ 2.33333, -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 - 7-3-2-11234-1010203040x = 2.3333333x = -1vertexy = 3x² - 4x - 7
Graph of y = 3x² - 4x - 7. The curve crosses the x-axis at x = 2.3333333 and x = -1; the vertex is at (0.66666667, -8.3333333).

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 = -4, c = -7.

    3x2−4x−7=03x^2 - 4x - 7 = 0
  2. 2

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

    D=(−4)2−4⋅3⋅(−7)=16+84=100D = (-4)^2 - 4\cdot3\cdot(-7) = 16 + 84 = 100
  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−7)(x+1)=0\left(3x - 7\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−7=0⇒x=73x+1=0⇒x=−13x - 7 = 0 \Rightarrow x = \frac{7}{3}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=4±1006x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{100}}{6}
  7. 7

    Simplify to get the solutions:

    x=73orx=−1x = \frac{7}{3}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 7/3 back in: a·x² + b·x + c = 3·(49/9) + (-4)·(7/3) + (-7) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 4x - 7 = 0\ \Rightarrow\ x = \frac{7}{3},\ -1 \]
\[ 3x^2 - 4x - 7 = 0 \]
\[ D = (-4)^2 - 4\cdot3\cdot(-7) = 16 + 84 = 100 \]
\[ \left(3x - 7\right)\left(x + 1\right) = 0 \]
\[ 3x - 7 = 0 \Rightarrow x = \frac{7}{3}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{100}}{6} \]
\[ x = \frac{7}{3}\quad\text{or}\quad x = -1 \]

The parabola y = 3x² - 4x - 7

xy
-2.333318.666067
-1.33333.6662667
-0.3333-5.3335333
0.6667-8.3333333
1.6667-5.3331333
2.66673.6670667
3.666718.667267

Questions

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

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

Does 3x² - 4x - 7 = 0 factor?

Yes: (3x-7)(x+1) = 0.

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

(2/3, -25/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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