x^2 - 6x + 10 = 0. How many solutions does the given equation have?

Step by step

The question

Question: how many solutions does x² − 6x + 10 = 0 have?

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Try factoring

First: can it split into two brackets?

We need two numbers that multiply to 10.

They must add to −6.

We list the pairs that make 10.

We add each pair.

None gives 6.

So it does not split into brackets with whole numbers.

2 Find a, b and c

So we use the formula.

It works for every equation ax² + bx + c = 0, with 0 on one side.

First we read the numbers.

a = 1 is with x².

b = −6 is with x.

c = 10 is the plain number.

3 Use the formula

Here is the formula.

People worked it out long ago, and the lesson on completing the square shows where it comes from.

± means we do it twice: once with plus, once with minus.

We put the numbers in, each in brackets.

−6 squared is 36.

4 times 1 times 10 is 40.

36 take away 40 is −4.

So under the root we have −4.

−4 is negative.

No number on the number line, times itself, is negative.

Those numbers are called real numbers.

So there is no real solution.

So the answer is: zero.

4 Pick the choice

No solution.

That means zero solutions.

That is choice A.

5 Why not the others

B is exactly one.

Wrong: one solution needs b² − 4ac = 0; here it is −4.

C is exactly two.

Wrong: two solutions need b² − 4ac above 0; here it is −4.

D is infinitely many.

Wrong: a quadratic has at most two solutions.

Let's look back.

First: try factoring.

Then: find a, b and c.

Then: use the formula.

Then: pick the choice.

Last: why not the others.