𝑥2 + 7𝑥 + 10 = 0
find the sum and product of roots
To find the sum and product of the roots of a quadratic equation, we need to use the quadratic formula.
The quadratic formula is given by:
𝑥 = (−𝑏 ± √(𝑏^2 − 4𝑎𝑐)) / 2𝑎
Where 𝑎, 𝑏, and 𝑐 are the coefficients of the quadratic equation 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0.
For the equation 𝑥^2 + 7𝑥 + 10 = 0, we can identify 𝑎 = 1, 𝑏 = 7, and 𝑐 = 10.
Plugging these values into the quadratic formula, we get:
𝑥 = (−7 ± √(7^2 − 4(1)(10))) / 2(1)
Simplifying further:
𝑥 = (−7 ± √(49 − 40)) / 2
𝑥 = (−7 ± √9) / 2
𝑥 = (−7 ± 3) / 2
Now, we can calculate the two possible roots:
For 𝑥 = (−7 + 3) / 2:
𝑥 = -4 / 2 = -2
For 𝑥 = (−7 - 3) / 2:
𝑥 = -10 / 2 = -5
Therefore, the roots of the equation 𝑥^2 + 7𝑥 + 10 = 0 are -2 and -5.
Now let's find the sum and product of these roots:
The sum of the roots is given by:
Sum = 𝑟1 + 𝑟2
Substituting the values 𝑟1 = -2 and 𝑟2 = -5:
Sum = -2 + (-5) = -7
The product of the roots is given by:
Product = 𝑟1 * 𝑟2
Substituting the values 𝑟1 = -2 and 𝑟2 = -5:
Product = -2 * (-5) = 10
Therefore, the sum of the roots is -7 and the product of the roots is 10 for the equation 𝑥^2 + 7𝑥 + 10 = 0.
To find the sum and product of the roots of the quadratic equation 𝑥^2 + 7𝑥 + 10 = 0, we can use the fact that if 𝑥₁ and 𝑥₂ are the roots of the quadratic equation 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0, then their sum is given by (-𝑏/𝑎) and their product is given by (𝑐/𝑎).
Comparing the given equation to the standard quadratic equation 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0, we have 𝑎 = 1, 𝑏 = 7, and 𝑐 = 10.
The sum of the roots is given by -𝑏/𝑎 = -7/1 = -7.
The product of the roots is given by 𝑐/𝑎 = 10/1 = 10.
Therefore, the sum of the roots is -7 and the product of the roots is 10.
recall that for ax^2 + bx + c
product of roots = c/a
sum = -b/a