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Solve -3x² - 7x - 4 = 0

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

x² +x += 0
−3x2−7x−4=0⇒x=−43, −1-3x^2 - 7x - 4 = 0 \Rightarrow x = -\frac{4}{3},\ -1
Discriminant D = 1 · x ≈ -1.33333, -1

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

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Graph of y = -3x² - 7x - 4-3-2-11-20-15-10-5x = -1.3333333x = -1vertexy = -3x² - 7x - 4
Graph of y = -3x² - 7x - 4. The curve crosses the x-axis at x = -1.3333333 and x = -1; the vertex is at (-1.1666667, 0.083333333).

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 = -4.

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

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

    D=(−7)2−4⋅(−3)⋅(−4)=49−48=1D = (-7)^2 - 4\cdot(-3)\cdot(-4) = 49 - 48 = 1
  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+4)(x+1)=0-\left(3x + 4\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+4=0⇒x=−43x+1=0⇒x=−13x + 4 = 0 \Rightarrow x = -\frac{4}{3}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=7±1−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{1}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−43orx=−1x = -\frac{4}{3}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = -4/3 back in: a·x² + b·x + c = -3·(16/9) + (-7)·(-4/3) + (-4) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 7x - 4 = 0\ \Rightarrow\ x = -\frac{4}{3},\ -1 \]
\[ -3x^2 - 7x - 4 = 0 \]
\[ D = (-7)^2 - 4\cdot(-3)\cdot(-4) = 49 - 48 = 1 \]
\[ -\left(3x + 4\right)\left(x + 1\right) = 0 \]
\[ 3x + 4 = 0 \Rightarrow x = -\frac{4}{3}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{1}}{-6} \]
\[ x = -\frac{4}{3}\quad\text{or}\quad x = -1 \]

The parabola y = -3x² - 7x - 4

xy
-4.1667-26.917267
-3.1667-11.917067
-2.1667-2.9168667
-1.16670.08333333
-0.1667-2.9164667
0.8333-11.916267
1.8333-26.916067

Questions

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

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

Does -3x² - 7x - 4 = 0 factor?

Yes: -1(3x+4)(x+1) = 0.

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

(-7/6, 1/12). 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 = 4/3.

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