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Solve 5x² + 5x - 10 = 0

The solutions of 5x² + 5x - 10 = 0 are x = 1 or x = -2.

x² +x += 0
5x2+5x−10=0⇒x=1, −25x^2 + 5x - 10 = 0 \Rightarrow x = 1,\ -2
Discriminant D = 225 · x ≈ 1, -2

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

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Graph of y = 5x² + 5x - 10-4-3-2-1123-20204060x = 1x = -2vertexy = 5x² + 5x - 10
Graph of y = 5x² + 5x - 10. The curve crosses the x-axis at x = 1 and x = -2; the vertex is at (-0.5, -11.25).

How to work it out

  1. 1

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

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

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

    D=52−4⋅5⋅(−10)=25+200=225D = 5^2 - 4\cdot5\cdot(-10) = 25 + 200 = 225
  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:

    5(x−1)(x+2)=05\left(x - 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:

    x−1=0⇒x=1x+2=0⇒x=−2x - 1 = 0 \Rightarrow x = 1\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−5±22510x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{225}}{10}
  7. 7

    Simplify to get the solutions:

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

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

LaTeX source of the formulas on this page
\[ 5x^2 + 5x - 10 = 0\ \Rightarrow\ x = 1,\ -2 \]
\[ 5x^2 + 5x - 10 = 0 \]
\[ D = 5^2 - 4\cdot5\cdot(-10) = 25 + 200 = 225 \]
\[ 5\left(x - 1\right)\left(x + 2\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{225}}{10} \]
\[ x = 1\quad\text{or}\quad x = -2 \]

The parabola y = 5x² + 5x - 10

xy
-3.533.75
-2.58.75
-1.5-6.25
-0.5-11.25
0.5-6.25
1.58.75
2.533.75

Questions

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

The solutions of 5x² + 5x - 10 = 0 are x = 1 or x = -2.

Does 5x² + 5x - 10 = 0 factor?

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

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

(-1/2, -45/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -1, product = c/a = -2.

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