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Solve 4x² + 10x + 4 = 0

The solutions of 4x² + 10x + 4 = 0 are x = -1/2 or x = -2.

x² +x += 0
4x2+10x+4=0⇒x=−12, −24x^2 + 10x + 4 = 0 \Rightarrow x = -\frac{1}{2},\ -2
Discriminant D = 36 · x ≈ -0.5, -2

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

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Graph of y = 4x² + 10x + 4-4-3-2-11210203040x = -0.5x = -2vertexy = 4x² + 10x + 4
Graph of y = 4x² + 10x + 4. The curve crosses the x-axis at x = -0.5 and x = -2; the vertex is at (-1.25, -2.25).

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 = 10, c = 4.

    4x2+10x+4=04x^2 + 10x + 4 = 0
  2. 2

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

    D=102−4⋅4⋅4=100−64=36D = 10^2 - 4\cdot4\cdot4 = 100 - 64 = 36
  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:

    2(2x+1)(x+2)=02\left(2x + 1\right)\left(x + 2\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=−12x+2=0⇒x=−22x + 1 = 0 \Rightarrow x = -\frac{1}{2}\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−10±368x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{36}}{8}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -1/2 back in: a·x² + b·x + c = 4·(1/4) + 10·(-1/2) + 4 = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 + 10x + 4 = 0\ \Rightarrow\ x = -\frac{1}{2},\ -2 \]
\[ 4x^2 + 10x + 4 = 0 \]
\[ D = 10^2 - 4\cdot4\cdot4 = 100 - 64 = 36 \]
\[ 2\left(2x + 1\right)\left(x + 2\right) = 0 \]
\[ 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{36}}{8} \]
\[ x = -\frac{1}{2}\quad\text{or}\quad x = -2 \]

The parabola y = 4x² + 10x + 4

xy
-4.2533.75
-3.2513.75
-2.251.75
-1.25-2.25
-0.251.75
0.7513.75
1.7533.75

Questions

What are the solutions of 4x² + 10x + 4 = 0?

The solutions of 4x² + 10x + 4 = 0 are x = -1/2 or x = -2.

Does 4x² + 10x + 4 = 0 factor?

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

What is the vertex of y = 4x² + 10x + 4?

(-5/4, -9/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5/2, product = c/a = 1.

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