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# Solving Quadratic Equations

## Converting Standard Form to Vertex Form

Completing the Square

You can covert any quadratic formula from standard form:

y = a Â· x 2 + b Â· x + c,

to vertex form:

y = a Â· (x - h) 2 + k,

through an algebraic process called completing the square. The next example demonstrates the steps that are involved in this process.

Example

Convert the quadratic function:

y = 4 Â· x 2 + 6 Â· x + 7,

from standard to vertex form and locate the x- and y-coordinates of the vertex.

Solution

Once the formula for the quadratic function has been converted to vertex form:

y = a Â· (x - h) 2 + k,

we can find the vertex by checking the vertex form to find the values of h (which will be the x-coordinate of the vertex) and k (which will be the y-coordinate of the vertex).

Conversion of the formula from standard to vertex form is a four-step process called completing the square.

1. Factor out the coefficient of x2 from all terms. 2. Add and subtract just the right amount1 to create a perfect square 1 To find just the right amount, you take the number that is left multiplying the x after Step 1 has been completed. Whatever this number is, divide the number by 2 and then take the square of what you are left with. This is just the right amount to create a perfect square.

3. Factor the perfect square and combine the constants 4. Distribute the factor that is out in front of the equation The vertex form of the quadratic function is: This is not quite the same as the â€œclassicâ€ format of the vertex form:

y = a Â· (x - h) 2 + k,

because the number inside the parentheses is added to x, rather than subtracted from x. To fix this we can use the fact that the negative of a negative is a positive, so that . Using this to re-write the vertex form in the â€œclassicâ€ format gives: With the vertex form in the â€œclassicâ€ format you can go ahead and determine the x- and y-coordinates of the vertex. The x-coordinate is and the y-coordinate is All Right Reserved. Copyright 2005-2022