factor x^2-15x+36 — step-by-step solution

Let's factorize in

ℝ

x^{2} - 15 x + 36

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -15

and

c = 36

Δ = {(-15)}^{2} - 4 \cdot 1 \cdot 36

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-15)}^{2}

(-15) \cdot (-15)

225

Δ = 225 - 4 \cdot 36

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 36

4 \cdot (30 + 6)

4 \cdot 30 + 4 \cdot 6

120 + 24

144

Δ = 225 - 144

ⓘ Start from

225

and move back

144

on the number line:

225 - 144

81

Δ = 81

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-15) - \sqrt{81}}{2 \cdot 1}

x_{1} = \frac{15 - \sqrt{81}}{2}

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

\sqrt{81}

9

;

9^{2}

81

x_{1} = \frac{15 - 9}{2}

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

15 - 9

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{- (-15) + \sqrt{81}}{2 \cdot 1}

x_{2} = \frac{15 + \sqrt{81}}{2}

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

\sqrt{81}

9

;

9^{2}

81

x_{2} = \frac{15 + 9}{2}

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

15 + 9

24

x_{2} = \frac{24}{2}

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

2

, their greatest common divisor:

\frac{24}{2}

12

x_{2} = 12

▶Factors identified:

(x - 3) \cdot (x - 12)