The equation y = x² - 4 represents a parabola that opens upwards because the coefficient of x² is positive. The vertex is at (0, -4) because the equation is in the form y = x² + c, where c is the vertical shift.
To solve the quadratic equation 2x² - 5x - 3 = 0, use the quadratic formula:
x = [-(-5) ± √((-5)² - 4(2)(-3))] / (2(2)).
Simplifying gives:
x = [5 ± √(25 + 24)] / 4
x = [5 ± √49] / 4
x = [5 ± 7] / 4.
Thus, x = (5 + 7)/4 = 12/4 = 3, or x = (5 - 7)/4 = -2/4 = -1/2.
Factor the numerator:
x² + 5x + 6 = (x + 2)(x + 3).
Now, cancel out the (x + 2) term in both the numerator and denominator:
(x + 2)(x + 3) ÷ (x + 2) = x + 3.
To find f(2), substitute 2 into the function:
f(2) = 2(2)³ - 5(2)² + 3
= 2(8) - 5(4) + 3
= 16 - 20 + 3
= 7.
Subtract 2x from both sides:
3x - 7 - 2x = 2x + 5 - 2x, so x - 7 = 5.
Next, add 7 to both sides:
x = 12.
