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Solve 4x² + 5x + 1 = 0

The solutions of 4x² + 5x + 1 = 0 are x = -1/4 or x = -1.

x² +x += 0
4x2+5x+1=0⇒x=−14, −14x^2 + 5x + 1 = 0 \Rightarrow x = -\frac{1}{4},\ -1
Discriminant D = 9 · x ≈ -0.25, -1

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

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Graph of y = 4x² + 5x + 1-3-2-1125101520253035x = -0.25x = -1vertexy = 4x² + 5x + 1
Graph of y = 4x² + 5x + 1. The curve crosses the x-axis at x = -0.25 and x = -1; the vertex is at (-0.625, -0.5625).

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 = 5, c = 1.

    4x2+5x+1=04x^2 + 5x + 1 = 0
  2. 2

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

    D=52−4⋅4⋅1=25−16=9D = 5^2 - 4\cdot4\cdot1 = 25 - 16 = 9
  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:

    (4x+1)(x+1)=0\left(4x + 1\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:

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

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−5±98x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{9}}{8}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -1/4 back in: a·x² + b·x + c = 4·(1/16) + 5·(-1/4) + 1 = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 + 5x + 1 = 0\ \Rightarrow\ x = -\frac{1}{4},\ -1 \]
\[ 4x^2 + 5x + 1 = 0 \]
\[ D = 5^2 - 4\cdot4\cdot1 = 25 - 16 = 9 \]
\[ \left(4x + 1\right)\left(x + 1\right) = 0 \]
\[ 4x + 1 = 0 \Rightarrow x = -\frac{1}{4}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{9}}{8} \]
\[ x = -\frac{1}{4}\quad\text{or}\quad x = -1 \]

The parabola y = 4x² + 5x + 1

xy
-3.62535.4375
-2.62515.4375
-1.6253.4375
-0.625-0.5625
0.3753.4375
1.37515.4375
2.37535.4375

Questions

What are the solutions of 4x² + 5x + 1 = 0?

The solutions of 4x² + 5x + 1 = 0 are x = -1/4 or x = -1.

Does 4x² + 5x + 1 = 0 factor?

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

What is the vertex of y = 4x² + 5x + 1?

(-5/8, -9/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5/4, product = c/a = 1/4.

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