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Solve 3x² + 4x - 7 = 0

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

x² +x += 0
3x2+4x−7=0⇒x=1, −733x^2 + 4x - 7 = 0 \Rightarrow x = 1,\ -\frac{7}{3}
Discriminant D = 100 · x ≈ 1, -2.33333

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

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Graph of y = 3x² + 4x - 7-4-3-2-1123-1010203040x = 1x = -2.3333333vertexy = 3x² + 4x - 7
Graph of y = 3x² + 4x - 7. The curve crosses the x-axis at x = 1 and x = -2.3333333; 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=42−4⋅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:

    (x−1)(3x+7)=0\left(x - 1\right)\left(3x + 7\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+7=0⇒x=−73x - 1 = 0 \Rightarrow x = 1\qquad 3x + 7 = 0 \Rightarrow x = -\frac{7}{3}
  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=1orx=−73x = 1\quad\text{or}\quad x = -\frac{7}{3}
  8. 8

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

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

The parabola y = 3x² + 4x - 7

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

Questions

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

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

Does 3x² + 4x - 7 = 0 factor?

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