Step by step
The question
Question: The height of a triangle is 6 cm more than its base.
The area is 56 square cm.
Find the base and the height.
1 Find what we know
What do we know?
The height is 6 more than the base.
The area is 56.
Area of a triangle = half of base times height.
What do we want?
The base and the height.
2 Name the unknown
We know the height from the base.
So we call the base x.
The height is x + 6.
Let's draw it.
The base is x.
The height is x + 6.
3 Write the equation
In words: half of base times height is the area.
So the equation is one half times x times (x + 6) = 56.
There is x times x in it: a quadratic equation.
Now we solve it.
First we remove the half: times 2 on both sides.
Half of something, times 2, is the whole thing.
56 times 2 is 112.
We open the bracket: x² + 6x = 112.
4 Get 0 on one side
To solve it, we want 0 on the right side.
So we take away 112 on both sides.
We do it on both sides, so they stay equal.
Now we have x² + 6x − 112 = 0.
5 Factor
We need two numbers.
They multiply to −112, and they add to 6.
They multiply to a minus number, so one number is plus and one is minus.
They add to a plus number, so the bigger one is plus.
We list the pairs that make 112, smallest first.
We take the smaller from the bigger.
We need 6.
8 and 14 give 6.
Yes!
With the signs: −8 and 14.
So it is (x − 8)(x + 14) = 0.
6 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
We make each bracket 0.
Two small equations.
In x − 8 = 0, we add 8 on both sides.
In x + 14 = 0, we take away 14 on both sides.
So x = 8 or x = −14.
7 Answer the question
x = −14: a length cannot be negative.
So we throw it away.
Careful!
Both numbers make the equation true.
But only one makes sense in the story.
Always look back at the story.
x = 8 works: a base.
The height is x + 6: 8 plus 6 is 14.
The base is 8 cm and the height is 14 cm.
8 Check
Let's check in the story.
Half of 8 times 14 is 56, the area.
And 14 is 8 plus 6, like the story says.
So the answer is right!
Let's look back.
First: find what we know.
Then: name the unknown.
Then: write the equation.
Then: solve it.
Then: answer the question.
Last: check.