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Solve 4x² - 3x - 1 = 0

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

x² +x += 0
4x2−3x−1=0⇒x=1, −144x^2 - 3x - 1 = 0 \Rightarrow x = 1,\ -\frac{1}{4}
Discriminant D = 25 · x ≈ 1, -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 - 3x - 1-2-112310203040x = 1x = -0.25vertexy = 4x² - 3x - 1
Graph of y = 4x² - 3x - 1. The curve crosses the x-axis at x = 1 and x = -0.25; the vertex is at (0.375, -1.5625).

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 = -3, c = -1.

    4x2−3x−1=04x^2 - 3x - 1 = 0
  2. 2

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

    D=(−3)2−4⋅4⋅(−1)=9+16=25D = (-3)^2 - 4\cdot4\cdot(-1) = 9 + 16 = 25
  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+1)=0\left(x - 1\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−1=0⇒x=14x+1=0⇒x=−14x - 1 = 0 \Rightarrow x = 1\qquad 4x + 1 = 0 \Rightarrow x = -\frac{1}{4}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=3±258x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{25}}{8}
  7. 7

    Simplify to get the solutions:

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

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

LaTeX source of the formulas on this page
\[ 4x^2 - 3x - 1 = 0\ \Rightarrow\ x = 1,\ -\frac{1}{4} \]
\[ 4x^2 - 3x - 1 = 0 \]
\[ D = (-3)^2 - 4\cdot4\cdot(-1) = 9 + 16 = 25 \]
\[ \left(x - 1\right)\left(4x + 1\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 4x + 1 = 0 \Rightarrow x = -\frac{1}{4} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{25}}{8} \]
\[ x = 1\quad\text{or}\quad x = -\frac{1}{4} \]

The parabola y = 4x² - 3x - 1

xy
-2.62534.4375
-1.62514.4375
-0.6252.4375
0.375-1.5625
1.3752.4375
2.37514.4375
3.37534.4375

Questions

What are the solutions of 4x² - 3x - 1 = 0?

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

Does 4x² - 3x - 1 = 0 factor?

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

What is the vertex of y = 4x² - 3x - 1?

(3/8, -25/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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