consider the following quadratic equations: x^(2)-7x+2=0 step 2 of 2: use the discriminant, b^(2)-4ac, to determine the number of solutions of the given quadratic equation, then solve the quadratic equation using the formula x=(-b+\sqrt(b^(2)-4ac))/(2a)
Step 1: Identify the coefficients of the quadratic equation:
In the equation x^(2)-7x+2=0,
a = 1
b = -7
c = 2
Step 2: Use the discriminant to determine the number of solutions:
The discriminant is given by Δ = b^(2) - 4ac
Plugging in the values, we get:
Δ = (-7)^(2) - 4(1)(2)
= 49 - 8
= 41
Since the discriminant (Δ) is greater than 0, the quadratic equation has two distinct real solutions.
Step 3: Solve the quadratic equation using the quadratic formula:
The quadratic formula is given by x = (-b ± √(b^(2) - 4ac))/(2a)
Plugging in the values, we get:
x = (-(-7) ± √((-7)^(2) - 4(1)(2))) / (2(1))
= (7 ± √(49 - 8)) / 2
= (7 ± √41) / 2
Therefore, the solutions to the quadratic equation x^(2)-7x+2=0 are:
x = (7 + √41) / 2
x = (7 - √41) / 2