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Solve x² - 3x - 10 = 0

The solutions of x² - 3x - 10 = 0 are x = 5 or x = -2.

x² +x += 0
x2−3x−10=0⇒x=5, −2x^2 - 3x - 10 = 0 \Rightarrow x = 5,\ -2
Discriminant D = 49 · x ≈ 5, -2

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

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Graph of y = x² - 3x - 10-4-2246-15-10-5510152025x = 5x = -2vertexy = x² - 3x - 10
Graph of y = x² - 3x - 10. The curve crosses the x-axis at x = 5 and x = -2; the vertex is at (1.5, -12.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = -3, c = -10.

    x2−3x−10=0x^2 - 3x - 10 = 0
  2. 2

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

    D=(−3)2−4⋅1⋅(−10)=9+40=49D = (-3)^2 - 4\cdot1\cdot(-10) = 9 + 40 = 49
  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:

    (x−5)(x+2)=0\left(x - 5\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−5=0⇒x=5x+2=0⇒x=−2x - 5 = 0 \Rightarrow x = 5\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=3±492x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{49}}{2}
  7. 7

    Simplify to get the solutions:

    x=5orx=−2x = 5\quad\text{or}\quad x = -2
  8. 8

    Check by putting x = 5 back in: a·x² + b·x + c = 1·(25) + (-3)·(5) + (-10) = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 - 3x - 10 = 0\ \Rightarrow\ x = 5,\ -2 \]
\[ x^2 - 3x - 10 = 0 \]
\[ D = (-3)^2 - 4\cdot1\cdot(-10) = 9 + 40 = 49 \]
\[ \left(x - 5\right)\left(x + 2\right) = 0 \]
\[ x - 5 = 0 \Rightarrow x = 5\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{49}}{2} \]
\[ x = 5\quad\text{or}\quad x = -2 \]

The parabola y = x² - 3x - 10

xy
-1.5-3.25
-0.5-8.25
0.5-11.25
1.5-12.25
2.5-11.25
3.5-8.25
4.5-3.25

Questions

What are the solutions of x² - 3x - 10 = 0?

The solutions of x² - 3x - 10 = 0 are x = 5 or x = -2.

Does x² - 3x - 10 = 0 factor?

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

What is the vertex of y = x² - 3x - 10?

(3/2, -49/4). 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 = -10.

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