Solve using the quadratic formula. Express answers as exact roots and as approximate roots, to the nearest hundredth.
b) k^2 - 9k = 1
c) 8w^2 = 2 - 3w
Solve to the nearest hundredth.
b) 1.2x^2 + 0.5x - 0.3 = 0
b. y = k^2 - 9k = 1.
Y = K^2 - 9K - 1 = 0,
K = (9 +- sqrt(81-(-4)) / 2=(9+-9.22)/2
= 9.11, and -0.11.
Use same procedure for remaining problems.
To solve the quadratic equations using the quadratic formula, we will use the general form of a quadratic equation: ax^2 + bx + c = 0. We can then plug in the coefficients from the given equations into the formula and solve for the variable.
b) For the equation k^2 - 9k = 1, we can rewrite it in the general form by moving 1 to the left-hand side: k^2 - 9k - 1 = 0. Comparing it to the general form, we have a = 1, b = -9, and c = -1. Now, we can apply the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values, we get:
k = (-(-9) ± √((-9)^2 - 4(1)(-1))) / (2(1))
k = (9 ± √(81 + 4)) / 2
k = (9 ± √85) / 2
Therefore, the exact roots are (9 + √85)/2 and (9 - √85)/2, and the approximate roots to the nearest hundredth are 7.79 and 1.21.
c) For the equation 8w^2 = 2 - 3w, we can rearrange it to have zero on the right-hand side: 8w^2 + 3w - 2 = 0. Comparing it to the general form, we have a = 8, b = 3, and c = -2. Now, we can apply the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values, we get:
w = (-(3) ± √((3)^2 - 4(8)(-2))) / (2(8))
w = (-3 ± √(9 + 64)) / 16
w = (-3 ± √73) / 16
Therefore, the exact roots are (-3 + √73)/16 and (-3 - √73)/16, and the approximate roots to the nearest hundredth are 0.18 and -0.39.
For the equation 1.2x^2 + 0.5x - 0.3 = 0, we already have it in general form. Comparing it to the general form, we have a = 1.2, b = 0.5, and c = -0.3. Now, we can apply the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values, we get:
x = (-(0.5) ± √((0.5)^2 - 4(1.2)(-0.3))) / (2(1.2))
x = (-0.5 ± √(0.25 + 1.44)) / 2.4
x = (-0.5 ± √1.69) / 2.4
Therefore, the exact roots are (-0.5 + √1.69)/2.4 and (-0.5 - √1.69)/2.4, and the approximate roots to the nearest hundredth are 0.21 and -0.96.