# The of a 15-foot ladder is 3 feet farther up

The of a 15-foot ladder is 3 feet farther up a wall than the foot is from the bottom of the wall. How far is the ladder from the bottom of the wall?

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Navreaiw
Step 1
Let:
$$\displaystyle{x}=\text{distance of the ladder from the bottom of the wall}$$
$$\displaystyle{y}=\text{height of the ladder from the bottom of the wall}$$
$$\displaystyle{L}=\text{length of the ladder}$$
Using Pythagorean theorem
$$\displaystyle{L}^{{{2}}}={x}^{{{2}}}+{y}^{{{2}}}$$
Since the height is 3 feet farther up than the distance:
$$\displaystyle{y}={x}+{3}$$
$$\displaystyle{L}^{{{2}}}={x}^{{{2}}}+{\left({x}+{3}\right)}^{{{2}}}$$
$$\displaystyle{\left({15}\right)}^{{{2}}}={x}^{{{2}}}+{x}^{{{2}}}+{6}{x}+{9}$$
$$\displaystyle{225}={2}{x}^{{{2}}}+{6}{x}+{9}$$
$$\displaystyle{2}{x}^{{{2}}}+{6}{x}-{216}={0}$$
$$\displaystyle{\left({x}-{9}\right)}{\left({x}+{12}\right)}={0}$$
$$\displaystyle{x}{1}={9}$$ and $$\displaystyle{x}{2}=-{12}$$ (neglect the negative value)
Answer: $$\displaystyle{x}={9}\text{feet}$$
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veiga34
Step 1
$$\displaystyle{15}^{{{2}}}={\left({x}+{3}\right)}^{{{2}}}+{x}^{{{2}}}$$
$$\displaystyle{225}={x}^{{{2}}}+{6}{x}+{9}+{x}^{{{2}}}$$
$$\displaystyle{2}{x}^{{{2}}}+{6}+{9}-{225}={0}$$
$$\displaystyle{2}{x}^{{{2}}}+{6}{x}-{216}={0}$$
$$\displaystyle{2}{\left({x}^{{{2}}}={3}{x}-{108}\right)}={0}$$
$$\displaystyle{2}{\left({X}={12}\right)}{\left({x}-{9}\right)}={0}$$
$$\displaystyle{x}-{9}={0}$$
$$\displaystyle{x}={9}$$ feet is the distance from the wall to the bottom of the ladder.
Proof:
$$\displaystyle{9}^{{{2}}}+{\left({9}+{3}\right)}^{{{2}}}={15}^{{{2}}}$$
$$\displaystyle{81}+{12}^{{{2}}}={225}$$
$$\displaystyle{81}+{144}={225}$$
$$\displaystyle{225}={225}$$
user_27qwe

Step 1
This is a right triangle with a hypotenuse =15, the base x & the other side =x+3
$$x^{2}+(x+3)^{2}=15^{2}$$
$$x^{2}+x^{2}+6x+9=225$$
$$2x^{2}+6x+9-225=0$$
$$2x^{2}+6x-216=0$$
$$2(X^{2}+3X-108)=0$$
$$2(X-9)(X+12)=0$$
$$X-9=0$$
X=9ft. Distance from the wall and the base of the ladder.
9+3=12ft. Distance of the ladder up the wall.