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Solve x² - 7x + 10 = 0

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

x² +x += 0
x2−7x+10=0⇒x=5, 2x^2 - 7x + 10 = 0 \Rightarrow x = 5,\ 2
Discriminant D = 9 · 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 - 7x + 101234567-4-22468101214x = 5x = 2vertexy = x² - 7x + 10
Graph of y = x² - 7x + 10. The curve crosses the x-axis at x = 5 and x = 2; 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=(−7)2−4⋅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−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=7±92x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{9}}{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) + (-7)·(5) + 10 = 0 ✓

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

The parabola y = x² - 7x + 10

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

Questions

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

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

Does x² - 7x + 10 = 0 factor?

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