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Solve -3x² + 7x + 6 = 0

The solutions of -3x² + 7x + 6 = 0 are x = -2/3 or x = 3.

x² +x += 0
−3x2+7x+6=0⇒x=−23, 3-3x^2 + 7x + 6 = 0 \Rightarrow x = -\frac{2}{3},\ 3
Discriminant D = 121 · x ≈ -0.666667, 3

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

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Graph of y = -3x² + 7x + 6-3-2-112345-50-40-30-20-1010x = -0.66666667x = 3vertexy = -3x² + 7x + 6
Graph of y = -3x² + 7x + 6. The curve crosses the x-axis at x = -0.66666667 and x = 3; the vertex is at (1.1666667, 10.083333).

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 = 6.

    −3x2+7x+6=0-3x^2 + 7x + 6 = 0
  2. 2

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

    D=72−4⋅(−3)⋅6=49+72=121D = 7^2 - 4\cdot(-3)\cdot6 = 49 + 72 = 121
  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+2)(x−3)=0-\left(3x + 2\right)\left(x - 3\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+2=0⇒x=−23x−3=0⇒x=33x + 2 = 0 \Rightarrow x = -\frac{2}{3}\qquad x - 3 = 0 \Rightarrow x = 3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−7±121−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{121}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−23orx=3x = -\frac{2}{3}\quad\text{or}\quad x = 3
  8. 8

    Check by putting x = -2/3 back in: a·x² + b·x + c = -3·(4/9) + 7·(-2/3) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 7x + 6 = 0\ \Rightarrow\ x = -\frac{2}{3},\ 3 \]
\[ -3x^2 + 7x + 6 = 0 \]
\[ D = 7^2 - 4\cdot(-3)\cdot6 = 49 + 72 = 121 \]
\[ -\left(3x + 2\right)\left(x - 3\right) = 0 \]
\[ 3x + 2 = 0 \Rightarrow x = -\frac{2}{3}\qquad x - 3 = 0 \Rightarrow x = 3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{121}}{-6} \]
\[ x = -\frac{2}{3}\quad\text{or}\quad x = 3 \]

The parabola y = -3x² + 7x + 6

xy
-1.8333-16.916067
-0.8333-1.9162667
0.16677.0835333
1.166710.083333
2.16677.0831333
3.1667-1.9170667
4.1667-16.917267

Questions

What are the solutions of -3x² + 7x + 6 = 0?

The solutions of -3x² + 7x + 6 = 0 are x = -2/3 or x = 3.

Does -3x² + 7x + 6 = 0 factor?

Yes: -1(3x+2)(x-3) = 0.

What is the vertex of y = -3x² + 7x + 6?

(7/6, 121/12). 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 = -2.

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