WordGPT

Solve -3x² - 11x + 4 = 0

The solutions of -3x² - 11x + 4 = 0 are x = -4 or x = 1/3.

x² +x += 0
−3x2−11x+4=0⇒x=−4, 13-3x^2 - 11x + 4 = 0 \Rightarrow x = -4,\ \frac{1}{3}
Discriminant D = 169 · x ≈ -4, 0.333333

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 - 11x + 4-6-5-4-3-2-112-50-40-30-20-101020x = -4x = 0.33333333vertexy = -3x² - 11x + 4
Graph of y = -3x² - 11x + 4. The curve crosses the x-axis at x = -4 and x = 0.33333333; the vertex is at (-1.8333333, 14.083333).

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 = -11, c = 4.

    −3x2−11x+4=0-3x^2 - 11x + 4 = 0
  2. 2

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

    D=(−11)2−4⋅(−3)⋅4=121+48=169D = (-11)^2 - 4\cdot(-3)\cdot4 = 121 + 48 = 169
  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+4)(3x−1)=0-\left(x + 4\right)\left(3x - 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:

    x+4=0⇒x=−43x−1=0⇒x=13x + 4 = 0 \Rightarrow x = -4\qquad 3x - 1 = 0 \Rightarrow x = \frac{1}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±169−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−4orx=13x = -4\quad\text{or}\quad x = \frac{1}{3}
  8. 8

    Check by putting x = -4 back in: a·x² + b·x + c = -3·(16) + (-11)·(-4) + 4 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 11x + 4 = 0\ \Rightarrow\ x = -4,\ \frac{1}{3} \]
\[ -3x^2 - 11x + 4 = 0 \]
\[ D = (-11)^2 - 4\cdot(-3)\cdot4 = 121 + 48 = 169 \]
\[ -\left(x + 4\right)\left(3x - 1\right) = 0 \]
\[ x + 4 = 0 \Rightarrow x = -4\qquad 3x - 1 = 0 \Rightarrow x = \frac{1}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{-6} \]
\[ x = -4\quad\text{or}\quad x = \frac{1}{3} \]

The parabola y = -3x² - 11x + 4

xy
-4.8333-12.916067
-3.83332.0837333
-2.833311.083533
-1.833314.083333
-0.833311.083133
0.16672.0829333
1.1667-12.917267

Questions

What are the solutions of -3x² - 11x + 4 = 0?

The solutions of -3x² - 11x + 4 = 0 are x = -4 or x = 1/3.

Does -3x² - 11x + 4 = 0 factor?

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

What is the vertex of y = -3x² - 11x + 4?

(-11/6, 169/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -11/3, product = c/a = -4/3.

Similar equations

    Solve -3x² - 11x + 4 = 0: x = -4 or x = 1/3 (steps) | WordGPT