Give the exact and approximate solutions to three decimal places.
X^2-7x+3=0
The exact solutions are x=________
The approximate solutions to 3decimal places are x=___________
[ 7 +/- sqrt (49 - 12) ] / 2
= 3.5 +/- .5 sqrt (37)
6.541
or
.459
To find the exact solutions of the quadratic equation x^2 - 7x + 3 = 0, we can use the quadratic formula:
x = [-b ± √(b^2 - 4ac)] / (2a)
Comparing the equation to the standard form ax^2 + bx + c = 0, we have a = 1, b = -7, and c = 3. Substituting these values into the quadratic formula, we get:
x = [-(-7) ± √((-7)^2 - 4(1)(3))] / (2(1))
x = [7 ± √(49 - 12)] / 2
x = [7 ± √37] / 2
Hence, the exact solutions are:
x = (7 + √37) / 2
x = (7 - √37) / 2
To find the approximate solutions to three decimal places, we can use a calculator to evaluate the square root of 37 and then divide the values accordingly. Evaluating √37 ≈ 6.083, we can substitute this into the equations:
x ≈ (7 + 6.083) / 2 ≈ 13.083 / 2 ≈ 6.542
x ≈ (7 - 6.083) / 2 ≈ 0.917 / 2 ≈ 0.458
Hence, the approximate solutions to three decimal places are:
x ≈ 6.542 and x ≈ 0.458.
To find the exact solutions to the equation x^2 - 7x + 3 = 0, we can use the quadratic formula. The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -7, and c = 3. Substituting these values into the quadratic formula, we get:
x = (-(-7) ± √((-7)^2 - 4(1)(3))) / (2(1))
Simplifying further:
x = (7 ± √(49 - 12)) / 2
x = (7 ± √37) / 2
Therefore, the exact solutions to the equation x^2 - 7x + 3 = 0 are:
x = (7 + √37) / 2
x = (7 - √37) / 2
To find the approximate solutions to three decimal places, we can plug the values of (7 ± √37) / 2 into a calculator:
For the positive solution:
x ≈ (7 + √37) / 2 ≈ 5.561
For the negative solution:
x ≈ (7 - √37) / 2 ≈ 1.439
Thus, the approximate solutions to three decimal places are:
x ≈ 5.561 and x ≈ 1.439.