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Solve 5x² + 10x + 5 = 0

The solution of 5x² + 10x + 5 = 0 is x = -1 (a repeated root).

x² +x += 0
5x2+10x+5=0⇒x=−15x^2 + 10x + 5 = 0 \Rightarrow x = -1
Discriminant D = 0 · x ≈ -1

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

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Graph of y = 5x² + 10x + 5-3.5-3-2.5-2-1.5-1-0.50.511.551015202530x = -1vertexy = 5x² + 10x + 5
Graph of y = 5x² + 10x + 5. The curve crosses the x-axis at x = -1; the vertex is at (-1, 0).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = 10, c = 5.

    5x2+10x+5=05x^2 + 10x + 5 = 0
  2. 2

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

    D=102−4⋅5⋅5=100−100=0D = 10^2 - 4\cdot5\cdot5 = 100 - 100 = 0
  3. 3

    D is zero, so there is exactly one solution (a repeated root).

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    5(x+1)(x+1)=05\left(x + 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:

    x+1=0⇒x=−1x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−10±010x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{0}}{10}
  7. 7

    Simplify to get the solution:

    x=−1x = -1
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = 5·(1) + 10·(-1) + 5 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 + 10x + 5 = 0\ \Rightarrow\ x = -1 \]
\[ 5x^2 + 10x + 5 = 0 \]
\[ D = 10^2 - 4\cdot5\cdot5 = 100 - 100 = 0 \]
\[ 5\left(x + 1\right)\left(x + 1\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{0}}{10} \]
\[ x = -1 \]

The parabola y = 5x² + 10x + 5

xy
-445
-320
-25
-10
05
120
245

Questions

What are the solutions of 5x² + 10x + 5 = 0?

The solution of 5x² + 10x + 5 = 0 is x = -1 (a repeated root).

Does 5x² + 10x + 5 = 0 factor?

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

What is the vertex of y = 5x² + 10x + 5?

(-1, 0). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -2, product = c/a = 1.

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