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Solve 4x² - x - 3 = 0

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

x² +x += 0
4x2−x−3=0⇒x=1, −344x^2 - x - 3 = 0 \Rightarrow x = 1,\ -\frac{3}{4}
Discriminant D = 49 · x ≈ 1, -0.75

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

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Graph of y = 4x² - x - 3-3-2-112310203040x = 1x = -0.75vertexy = 4x² - x - 3
Graph of y = 4x² - x - 3. The curve crosses the x-axis at x = 1 and x = -0.75; the vertex is at (0.125, -3.0625).

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 = -1, c = -3.

    4x2−x−3=04x^2 - x - 3 = 0
  2. 2

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

    D=(−1)2−4⋅4⋅(−3)=1+48=49D = (-1)^2 - 4\cdot4\cdot(-3) = 1 + 48 = 49
  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)(4x+3)=0\left(x - 1\right)\left(4x + 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:

    x−1=0⇒x=14x+3=0⇒x=−34x - 1 = 0 \Rightarrow x = 1\qquad 4x + 3 = 0 \Rightarrow x = -\frac{3}{4}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±498x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{49}}{8}
  7. 7

    Simplify to get the solutions:

    x=1orx=−34x = 1\quad\text{or}\quad x = -\frac{3}{4}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 4·(1) + (-1)·(1) + (-3) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - x - 3 = 0\ \Rightarrow\ x = 1,\ -\frac{3}{4} \]
\[ 4x^2 - x - 3 = 0 \]
\[ D = (-1)^2 - 4\cdot4\cdot(-3) = 1 + 48 = 49 \]
\[ \left(x - 1\right)\left(4x + 3\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 4x + 3 = 0 \Rightarrow x = -\frac{3}{4} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{49}}{8} \]
\[ x = 1\quad\text{or}\quad x = -\frac{3}{4} \]

The parabola y = 4x² - x - 3

xy
-2.87532.9375
-1.87512.9375
-0.8750.9375
0.125-3.0625
1.1250.9375
2.12512.9375
3.12532.9375

Questions

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

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

Does 4x² - x - 3 = 0 factor?

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

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

(1/8, -49/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1/4, product = c/a = -3/4.

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