Quadratic Equations

Solve for x: 2^2x – 8(2^x) + 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 2^x.

2^2x – 8(2^x) + 15 = 0

(2^x)² – 8(2^x) + 15 = 0

So if we let y = 2^x, then (2^x)² – 8(2^x) + 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 = 2^x, then 2^x = 5, or 2^x = 3.

To solve when x is the exponent we use from the rules of indices and logarithms the fact that

a^x = y <=> log_ay = x

2^x = 5 <=> log₂5 = x = 1.61 to two decimal places

2^x = 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.

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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