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Solve x² + 7x + 10 = 0

The solutions of x² + 7x + 10 = 0 are x = -2 or x = -5.

x² +x += 0
x2+7x+10=0⇒x=−2, −5x^2 + 7x + 10 = 0 \Rightarrow x = -2,\ -5
Discriminant D = 9 · 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 + 7x + 10-7-6-5-4-3-2-1-4-22468101214x = -2x = -5vertexy = x² + 7x + 10
Graph of y = x² + 7x + 10. The curve crosses the x-axis at x = -2 and x = -5; the vertex is at (-3.5, -2.25).

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 = 7, c = 10.

    x2+7x+10=0x^2 + 7x + 10 = 0
  2. 2

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

    D=72−4⋅1⋅10=49−40=9D = 7^2 - 4\cdot1\cdot10 = 49 - 40 = 9
  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=−7±92x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{9}}{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) + 7·(-2) + 10 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 + 7x + 10 = 0\ \Rightarrow\ x = -2,\ -5 \]
\[ x^2 + 7x + 10 = 0 \]
\[ D = 7^2 - 4\cdot1\cdot10 = 49 - 40 = 9 \]
\[ \left(x + 2\right)\left(x + 5\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad x + 5 = 0 \Rightarrow x = -5 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{9}}{2} \]
\[ x = -2\quad\text{or}\quad x = -5 \]

The parabola y = x² + 7x + 10

xy
-6.56.75
-5.51.75
-4.5-1.25
-3.5-2.25
-2.5-1.25
-1.51.75
-0.56.75

Questions

What are the solutions of x² + 7x + 10 = 0?

The solutions of x² + 7x + 10 = 0 are x = -2 or x = -5.

Does x² + 7x + 10 = 0 factor?

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

What is the vertex of y = x² + 7x + 10?

(-7/2, -9/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -7, product = c/a = 10.

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