WordGPT

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, −33x^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.

Explain it step by step in chatMake a short video of it
Graph of y = 3x² + 7x - 6-5-4-3-2-1123-101020304050x = 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=03x^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\cdot3\cdot(-6) = 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±1216x = \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
\[ 3x^2 + 7x - 6 = 0\ \Rightarrow\ x = \frac{2}{3},\ -3 \]
\[ 3x^2 + 7x - 6 = 0 \]
\[ D = 7^2 - 4\cdot3\cdot(-6) = 49 + 72 = 121 \]
\[ \left(3x - 2\right)\left(x + 3\right) = 0 \]
\[ 3x - 2 = 0 \Rightarrow x = \frac{2}{3}\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{121}}{6} \]
\[ x = \frac{2}{3}\quad\text{or}\quad x = -3 \]

The parabola y = 3x² + 7x - 6

xy
-4.166716.917267
-3.16671.9170667
-2.1667-7.0831333
-1.1667-10.083333
-0.1667-7.0835333
0.83331.9162667
1.833316.916067

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

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

(-7/6, -121/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -7/3, product = c/a = -2.

Similar equations

    Solve 3x² + 7x - 6 = 0: x = 2/3 or x = -3 (steps) | WordGPT