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Solve 4x² + 12x + 5 = 0

The solutions of 4x² + 12x + 5 = 0 are x = -1/2 or x = -5/2.

x² +x += 0
4x2+12x+5=0⇒x=−12, −524x^2 + 12x + 5 = 0 \Rightarrow x = -\frac{1}{2},\ -\frac{5}{2}
Discriminant D = 64 · x ≈ -0.5, -2.5

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

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Graph of y = 4x² + 12x + 5-5-4-3-2-112-1010203040x = -0.5x = -2.5vertexy = 4x² + 12x + 5
Graph of y = 4x² + 12x + 5. The curve crosses the x-axis at x = -0.5 and x = -2.5; the vertex is at (-1.5, -4).

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 = 12, c = 5.

    4x2+12x+5=04x^2 + 12x + 5 = 0
  2. 2

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

    D=122−4⋅4⋅5=144−80=64D = 12^2 - 4\cdot4\cdot5 = 144 - 80 = 64
  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:

    (2x+1)(2x+5)=0\left(2x + 1\right)\left(2x + 5\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    2x+1=0⇒x=−122x+5=0⇒x=−522x + 1 = 0 \Rightarrow x = -\frac{1}{2}\qquad 2x + 5 = 0 \Rightarrow x = -\frac{5}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−12±648x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{64}}{8}
  7. 7

    Simplify to get the solutions:

    x=−12orx=−52x = -\frac{1}{2}\quad\text{or}\quad x = -\frac{5}{2}
  8. 8

    Check by putting x = -1/2 back in: a·x² + b·x + c = 4·(1/4) + 12·(-1/2) + 5 = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 + 12x + 5 = 0\ \Rightarrow\ x = -\frac{1}{2},\ -\frac{5}{2} \]
\[ 4x^2 + 12x + 5 = 0 \]
\[ D = 12^2 - 4\cdot4\cdot5 = 144 - 80 = 64 \]
\[ \left(2x + 1\right)\left(2x + 5\right) = 0 \]
\[ 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}\qquad 2x + 5 = 0 \Rightarrow x = -\frac{5}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{64}}{8} \]
\[ x = -\frac{1}{2}\quad\text{or}\quad x = -\frac{5}{2} \]

The parabola y = 4x² + 12x + 5

xy
-4.532
-3.512
-2.50
-1.5-4
-0.50
0.512
1.532

Questions

What are the solutions of 4x² + 12x + 5 = 0?

The solutions of 4x² + 12x + 5 = 0 are x = -1/2 or x = -5/2.

Does 4x² + 12x + 5 = 0 factor?

Yes: (2x+1)(2x+5) = 0.

What is the vertex of y = 4x² + 12x + 5?

(-3/2, -4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = 5/4.

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