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Solve 5x² - 5x - 10 = 0

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

x² +x += 0
5x2−5x−10=0⇒x=2, −15x^2 - 5x - 10 = 0 \Rightarrow x = 2,\ -1
Discriminant D = 225 · x ≈ 2, -1

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

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Graph of y = 5x² - 5x - 10-3-2-11234-20204060x = 2x = -1vertexy = 5x² - 5x - 10
Graph of y = 5x² - 5x - 10. The curve crosses the x-axis at x = 2 and x = -1; 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=(−5)2−4⋅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−2)(x+1)=05\left(x - 2\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−2=0⇒x=2x+1=0⇒x=−1x - 2 = 0 \Rightarrow x = 2\qquad x + 1 = 0 \Rightarrow x = -1
  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=2orx=−1x = 2\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = 5·(4) + (-5)·(2) + (-10) = 0 ✓

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

The parabola y = 5x² - 5x - 10

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

Questions

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

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

Does 5x² - 5x - 10 = 0 factor?

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