Final answer:
To solve 3x^2+7=31 using square roots, subtract 7 from both sides, divide by 3, and take the square root. To solve x^2-12x=45 by factoring, rearrange the equation, factor, and solve for x. To find the zeros of f(x) = 6x^2+9x-6, use the quadratic formula and simplify.
Step-by-step explanation:
Solving 3x^2+7=31 using square roots:
First, subtract 7 from both sides of the equation:
3x^2 = 24
Next, divide both sides by 3 to isolate x^2:
x^2 = 8
Finally, take the square root of both sides to solve for x:
x = ±√8
Simplifying, we get x = ±2√2
Solving x^2-12x=45 by factoring:
Begin by rearranging the equation:
x^2 - 12x - 45 = 0
Factor the quadratic equation into two binomials:
(x - 15)(x + 3) = 0
Set each binomial equal to zero and solve for x:
x - 15 = 0 or x + 3 = 0
Solving for x, we get x = 15 or x = -3
Finding the zeros of f(x) = 6x^2+9x-6:
Set f(x) equal to zero:
6x^2 + 9x - 6 = 0
Use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values from the quadratic equation into the formula:
x = (-9 ± √(9^2 - 4 * 6 * -6)) / (2 * 6)
Simplifying, we get x = (-9 ± √(81 + 144)) / 12
x = (-9 ± √(225)) / 12
x = (-9 ± 15) / 12
Solving for x, we get x = 1/2 or x = -3