solve x^2-10x+21=0 — step-by-step solution

Let's solve in

ℝ

x^{2} - 10 x + 21 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -10

and

c = 21

Δ = {(-10)}^{2} - 4 \cdot 1 \cdot 21

ⓘ Powers are computed before multiplications, additions and subtractions. A power with a positive integer exponent is a repeated product:

{(-10)}^{2}

(-10) \cdot (-10)

100

Δ = 100 - 4 \cdot 21

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 21

4 \cdot (20 + 1)

4 \cdot 20 + 4

80 + 4

84

Δ = 100 - 84

ⓘ Start from

100

and move back

84

on the number line:

100 - 84

16

Δ = 16

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-10) - \sqrt{16}}{2 \cdot 1}

x_{1} = \frac{10 - \sqrt{16}}{2}

ⓘ The square root of a nonnegative real number is the nonnegative number whose square is that number:

\sqrt{16}

4

since

4^{2}

16

x_{1} = \frac{10 - 4}{2}

ⓘ The numerator and the denominator are computed before the division:

10 - 4

6

x_{1} = \frac{6}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{6}{2}

3

x_{1} = 3

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-10) + \sqrt{16}}{2 \cdot 1}

x_{2} = \frac{10 + \sqrt{16}}{2}

ⓘ The square root of a nonnegative real number is the nonnegative number whose square is that number:

\sqrt{16}

4

because

4^{2}

16

x_{2} = \frac{10 + 4}{2}

ⓘ The numerator and the denominator are computed before the division:

10 + 4

14

x_{2} = \frac{14}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{14}{2}

7

x_{2} = 7

▶Solutions obtained:

x = {3, 7}