7x^2-14x+5=-x^2 — step-by-step solution

1) Let's solve in

ℝ

7 x^{2} - 14 x + 5 = - x^{2}

7 x^{2}

- 14 x

+ 5

+ x^{2} = 0

ⓘ Like terms are grouped by adding their coefficients:

7 x^{2} + x^{2}

8 x^{2}

8 x^{2} - 14 x + 5 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 8

b = -14

and

c = 5

Δ = {(-14)}^{2} - 4 \cdot 8 \cdot 5

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

{(-14)}^{2}

(-14) \cdot (-14)

196

■Multiplications are done before additions and subtractions:

4 \cdot 8

32

Δ = 196 - 32 \cdot 5

ⓘ Multiplications are done before additions and subtractions:

32 \cdot 5

(30 + 2) \cdot 5

30 \cdot 5 + 2 \cdot 5

150 + 10

160

Δ = 196 - 160

ⓘ Start from

196

and move back

160

on the number line:

196 - 160

36

Δ = 36

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-14) - \sqrt{36}}{2 \cdot 8}

x_{1} = \frac{14 - \sqrt{36}}{2 \cdot 8}

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

\sqrt{36}

6

;

6^{2}

36

ⅰ

2 \cdot 8

16

x_{1} = \frac{14 - 6}{16}

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

14 - 6

8

x_{1} = \frac{8}{16}

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

8

, their greatest common divisor:

\frac{8}{16}

\frac{1 \cdot 8}{2 \cdot 8}

\frac{1}{2}

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-14) + \sqrt{36}}{2 \cdot 8}

x_{2} = \frac{14 + \sqrt{36}}{2 \cdot 8}

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

\sqrt{36}

6

;

6^{2}

36

ⅰ

2 \cdot 8

16

x_{2} = \frac{14 + 6}{16}

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

14 + 6

20

x_{2} = \frac{20}{16}

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

4

, their greatest common divisor:

\frac{20}{16}

\frac{5 \cdot 4}{4 \cdot 4}

\frac{5}{4}

x_{2} = \frac{5}{4}

▶Solutions found:

x = {\frac{1}{2}, \frac{5}{4}}

2) The numeric value of:

x = {\frac{1}{2}, \frac{5}{4}}

equals:

▷

x = {0.5, 1.25}