How do you find the roots of the equation x² + 7x + 10 = 0?

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To find the roots of the quadratic equation x² + 7x + 10 = 0, we can use the quadratic formula, factorization, or completing the square. Here, we’ll explore both factorization and the quadratic formula methods.

Method 1: Factorization

Step 1: Rewrite the equation in standard form:

x² + 7x + 10 = 0

Step 2: Look for two numbers that multiply to 10 (the constant term) and add to 7 (the coefficient of x). The numbers that satisfy this are 2 and 5.

Step 3: Rewrite the quadratic equation in factored form:

(x + 2)(x + 5) = 0

Step 4: Set each factor equal to zero and solve for x:

x + 2 = 0  or  x + 5 = 0

This gives us:

x = -2  or  x = -5

Method 2: Quadratic Formula

Alternatively, we can use the quadratic formula, which is:

x = (-b ± √(b² - 4ac)) / 2a

In our case, a = 1, b = 7, and c = 10.

Step 1: Calculate the discriminant:

b² - 4ac = 7² - 4(1)(10) = 49 - 40 = 9

Step 2: Plug the values into the quadratic formula:

x = ( -7 ± √9 ) / 2(1) = ( -7 ± 3 ) / 2

Step 3: Solve for the two possible values of x:

  • x = (-7 + 3) / 2 = -4 / 2 = -2
  • x = (-7 – 3) / 2 = -10 / 2 = -5

So the roots of the equation x² + 7x + 10 = 0 are:

  • x = -2
  • x = -5

In conclusion, whether you approach it through factorization or the quadratic formula, you will find that the roots of the equation are -2 and -5.

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