Can some one help me understand this question please? Give exact and approximate solutions to three decimal places. x^2-7x+11=0
Sure! The given question is asking for the exact and approximate solutions to the equation x^2 - 7x + 11 = 0, with both solutions rounded to three decimal places.
To solve this equation, we can use the quadratic formula, which states that for any quadratic equation in the form ax^2 + bx + c = 0, the solutions can be found using x = (-b ± √(b^2 - 4ac)) / (2a).
Comparing our equation to the general form, we have:
a = 1, b = -7, c = 11
Now we can substitute these values into the quadratic formula and solve for x:
x = (-(-7) ± √((-7)^2 - 4 * 1 * 11)) / (2 * 1)
= (7 ± √(49 - 44)) / 2
= (7 ± √(5)) / 2
So the exact solutions to the equation are:
x₁ = (7 + √(5)) / 2
x₂ = (7 - √(5)) / 2
Now, let's calculate the approximate solutions rounded to three decimal places. We can use a calculator to find the approximate values of √(5) and perform the calculations:
Approximate value of √(5) ≈ 2.236
For x₁:
x₁ ≈ (7 + 2.236) / 2
≈ 4.618
For x₂:
x₂ ≈ (7 - 2.236) / 2
≈ 1.382
Therefore, the exact solutions to the equation x^2 - 7x + 11 = 0 are x₁ = (7 + √(5)) / 2 and x₂ = (7 - √(5)) / 2. And the approximate solutions rounded to three decimal places are x₁ ≈ 4.618 and x₂ ≈ 1.382.