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Solve x² + 3x - 10 = 0

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

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

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

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Graph of y = x² + 3x - 10-6-4-224-15-10-5510152025x = 2x = -5vertexy = x² + 3x - 10
Graph of y = x² + 3x - 10. The curve crosses the x-axis at x = 2 and x = -5; 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=32−4⋅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−2)(x+5)=0\left(x - 2\right)\left(x + 5\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+5=0⇒x=−5x - 2 = 0 \Rightarrow x = 2\qquad x + 5 = 0 \Rightarrow x = -5
  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=2orx=−5x = 2\quad\text{or}\quad x = -5
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = 1·(4) + 3·(2) + (-10) = 0 ✓

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

The parabola y = x² + 3x - 10

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

Questions

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

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

Does x² + 3x - 10 = 0 factor?

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