How many solutions x^2 - 3x - 10 = 0 Step by Step

Step by step

The question

Question: find the number of real solutions of x² − 3x − 10 = 0.

x² − 3x − 10 = 0

1 Find a, b and c

We read the numbers: a = 1 is with x², b = −3 is with x, c = −10 is the plain number.

a = 1, b = −3, c = −10

2 Work out b² − 4ac

In the formula, b² − 4ac is under the square root.

It is called the discriminant.

Its sign tells us how many solutions there are.

x = −b ± √b² − 4ac2a

We put the numbers in, each negative one in brackets.

b² − 4ac = (−3)² − 4 × 1 × (−10)

(−3)² is 9.

Minus times minus is plus.

4 times 1 times −10 is −40.

(−3)² = (−3) × (−3) = 9

4 × 1 × (−10) = −40

Careful!

b = −3 goes in brackets.

(−3) squared is 9, not minus 9.

Careful: not −3² = −9 ✗

So we have 9 take away −40.

= 9 − (−40)

Taking away a negative is the same as adding.

9 − (−40) = 9 + 40

So the discriminant is 49.

= 49

3 How many solutions

49 is positive.

The square root of a positive number is a real number.

The plus or minus gives two different answers.

positive → 2 real solutions

0 → 1 real solution

negative → no real solutions

So x² − 3x − 10 = 0 has 2 real solutions.

= 2 real solutions

4 Check

Let's check with the formula.

Under the root we get 49.

Its square root is 7.

x = 3 ± 72

The plus gives 5, the minus gives −2.

Two different solutions.

It works!

x = 5 or −2

Let's look back.

First: find a, b and c.

Then: work out b² − 4ac.

Then: how many solutions.

Last: check.

Answer