**Word problems**

The applications of quadratic equations can be referred to word problems from which a quadratic equation is formed.

A lot of parabolas are used in construction, for example, the arches of a bridge can be parabola-shaped.

**Example 1**

Liisa-Petter ordered a bridge online so she could visit her cousin Klaus-Heidi on the neighbouring island.

On a Chinese online store, bridges were marked with parabolas, and the unit of measure was in metres. Which bridge should Liisa-Petter order when a 30-metre-long bridge is needed between the islands?

The unit of measurement in the store is in metres, so the distance between the zeros is the length of the bridge. The zeros of the first bridge are* x = 0* and* x = 30*, the zeros of the second are* x = 0* and *x = 32*, and the zeros of the third are* x = 0* and *x = 20*.

Liisa-Petter should choose the first bridge.

However, Klaus-Heidi was scared of going on tall bridges, so Liisa-Petter had to work out the height of the bridge.

The highest point of the bridge is at the top of the parabola. The peak, is halfway between the zero points, that is, when *x = 15.*

In this case, the height is *4,5* metres

**Example 2**

Liisa-Petter had a plot of land *80 *metres wide and *120* metres long. She bought some additional space to her plot so that the width and length would increase equally. The area of the new plot is *11700 **m*^{2} . What are the dimensions of the new plot?

We concept the situation and mark the increase with *x.* The area of the new plot is then *(80 + x) (120 + x) = 11700*. Simplified, this is

the quadratic equation has *a = 1, b = 200*, and *c = -2100*. Substitute into the formula

The solutions are* x = -210 or x = 10*. The new dimensions are therefore *90* metres wide and *130 *metres long.

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