Solve for x: 22x – 8(2x) + 15 = 0. Give your answer correct to two decimal places.
This is easier than might first appear if we realise that it is a quadratic equation in terms of 2x.
22x – 8(2x) + 15 = 0
(2x)2 – 8(2x) + 15 = 0
So if we let y = 2x, then (2x)2 – 8(2x) + 15 = 0 can be rewritten as y2 – 8y + 15 = 0.
We can use factorisation or our “minus b” formula to solve for y.
Using factorisation: (y-5)(y-3) = 0
y = 5 or y = 3
But since y = 2x, then 2x = 5, or 2x = 3.
To solve when x is the exponent we use from the rules of indices and logarithms the fact that
ax = y <=> logay = x
2x = 5 <=> log25 = x = 1.61 to two decimal places
2x = 3 <=> log23 = x = 1.10 to two decimal places
Vertex Form of a Quadratic Equation
The vertex form of a quadratic equation is given by y = a(x – h)2 + k, where (h, k) is the vertex of the parabola.
The h represents the horizontal shift. How far left or right the graph is shifted from x = 0 from the parent equation y = x2.
And k represents the vertical shift. How far up or down the graph is shifted from y = 0 from the parent equation y = x2.
Example: Sketch a graph of y = 5 – 3(x+2)2.
Comparing y = 5 – 3(x+2)2 with y = a(x – h)2 + k, we get (h,k) = (-2,5) for the vertex of the parabola.
To Convert from Vertex Form to y = ax2 + bx + c Form:
Simply multiply out and combine like terms:
y = 5 – 3(x+2)2
= 5 – 3(x2+4x+4)
= 5- 3x2-12x-12
= – 3x2-12x-7
Using minus B formula, roots are: x = -3.291 or x = -0.709