2x^2-13x+15=0 — step-by-step solution

1) Let's solve in

ℝ

2 x^{2} - 13 x + 15 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = -13

and

c = 15

Δ = {(-13)}^{2} - 4 \cdot 2 \cdot 15

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

{(-13)}^{2}

(-13) \cdot (-13)

169

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

Δ = 169 - 8 \cdot 15

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 15

8 \cdot (10 + 5)

8 \cdot 10 + 8 \cdot 5

80 + 40

120

Δ = 169 - 120

ⓘ Start from

169

and move back

120

on the number line:

169 - 120

49

Δ = 49

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-13) - \sqrt{49}}{2 \cdot 2}

x_{1} = \frac{13 - \sqrt{49}}{2 \cdot 2}

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

\sqrt{49}

7

;

7^{2}

49

ⅰ

2 \cdot 2

4

x_{1} = \frac{13 - 7}{4}

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

13 - 7

6

x_{1} = \frac{6}{4}

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

2

, their greatest common divisor:

\frac{6}{4}

\frac{3 \cdot 2}{2 \cdot 2}

\frac{3}{2}

x_{1} = \frac{3}{2}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-13) + \sqrt{49}}{2 \cdot 2}

x_{2} = \frac{13 + \sqrt{49}}{2 \cdot 2}

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

\sqrt{49}

7

;

7^{2}

49

ⅰ

2 \cdot 2

4

x_{2} = \frac{13 + 7}{4}

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

13 + 7

20

x_{2} = \frac{20}{4}

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

4

, their greatest common divisor:

\frac{20}{4}

5

x_{2} = 5

▶Solutions found:

x = {\frac{3}{2}, 5}

2) The numeric value of:

x = {\frac{3}{2}, 5}

is:

▷

x = {1.5, 5}