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Solve 4x² - 11x - 3 = 0

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

x² +x += 0
4x2−11x−3=0⇒x=3, −144x^2 - 11x - 3 = 0 \Rightarrow x = 3,\ -\frac{1}{4}
Discriminant D = 169 · x ≈ 3, -0.25

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

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Graph of y = 4x² - 11x - 3-2-112345-10102030405060x = 3x = -0.25vertexy = 4x² - 11x - 3
Graph of y = 4x² - 11x - 3. The curve crosses the x-axis at x = 3 and x = -0.25; the vertex is at (1.375, -10.5625).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = -11, c = -3.

    4x2−11x−3=04x^2 - 11x - 3 = 0
  2. 2

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

    D=(−11)2−4⋅4⋅(−3)=121+48=169D = (-11)^2 - 4\cdot4\cdot(-3) = 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−3)(4x+1)=0\left(x - 3\right)\left(4x + 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−3=0⇒x=34x+1=0⇒x=−14x - 3 = 0 \Rightarrow x = 3\qquad 4x + 1 = 0 \Rightarrow x = -\frac{1}{4}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±1698x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{8}
  7. 7

    Simplify to get the solutions:

    x=3orx=−14x = 3\quad\text{or}\quad x = -\frac{1}{4}
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = 4·(9) + (-11)·(3) + (-3) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 11x - 3 = 0\ \Rightarrow\ x = 3,\ -\frac{1}{4} \]
\[ 4x^2 - 11x - 3 = 0 \]
\[ D = (-11)^2 - 4\cdot4\cdot(-3) = 121 + 48 = 169 \]
\[ \left(x - 3\right)\left(4x + 1\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad 4x + 1 = 0 \Rightarrow x = -\frac{1}{4} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{8} \]
\[ x = 3\quad\text{or}\quad x = -\frac{1}{4} \]

The parabola y = 4x² - 11x - 3

xy
-1.62525.4375
-0.6255.4375
0.375-6.5625
1.375-10.5625
2.375-6.5625
3.3755.4375
4.37525.4375

Questions

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

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

Does 4x² - 11x - 3 = 0 factor?

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

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

(11/8, -169/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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