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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, 73-3x^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 + 7-3-2-11234-40-30-20-1010x = -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=0-3x^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\cdot(-3)\cdot7 = 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±100−6x = \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
\[ -3x^2 + 4x + 7 = 0\ \Rightarrow\ x = -1,\ \frac{7}{3} \]
\[ -3x^2 + 4x + 7 = 0 \]
\[ D = 4^2 - 4\cdot(-3)\cdot7 = 16 + 84 = 100 \]
\[ -\left(x + 1\right)\left(3x - 7\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 3x - 7 = 0 \Rightarrow x = \frac{7}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{100}}{-6} \]
\[ x = -1\quad\text{or}\quad x = \frac{7}{3} \]

The parabola y = -3x² + 4x + 7

xy
-2.3333-18.666067
-1.3333-3.6662667
-0.33335.3335333
0.66678.3333333
1.66675.3331333
2.6667-3.6670667
3.6667-18.667267

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

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

(2/3, 25/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

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

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