-16t^2+6t+5=0 — step-by-step solution

1) Let's solve in

ℝ

- 16 t^{2} + 6 t + 5 = 0

Since

-1 \neq 0

, divide both sides by

-1

16 t^{2} - 6 t - 5 = 0

Given the quadratic form

a t^{2} + b t + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 16

b = -6

and

c = -5

Δ = {(-6)}^{2} - 4 \cdot 16 \cdot (-5)

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

{(-6)}^{2}

(-6) \cdot (-6)

36

■Multiplications are done before additions and subtractions:

4 \cdot 16

4 \cdot (10 + 6)

4 \cdot 10 + 4 \cdot 6

40 + 24

64

■Two minus signs cancel out, the product of two negative numbers is positive:

- 64 \cdot (-5)

64 \cdot 5

Δ = 36 + 64 \cdot 5

ⓘ Multiplications are done before additions and subtractions:

64 \cdot 5

(60 + 4) \cdot 5

60 \cdot 5 + 4 \cdot 5

300 + 20

320

Δ = 36 + 320

ⓘ Start from

36

and move forward

320

on the number line:

36 + 320

356

Δ = 356

Δ > 0

, the equation has two roots:

The quadratic formula gives

t_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-6) - \sqrt{356}}{2 \cdot 16}

t_{1} = \frac{6 - \sqrt{356}}{2 \cdot 16}

ⓘ We look for the largest square dividing

356

4

2^{2}

, and take it out of the root:

\sqrt{356}

\sqrt{4 \cdot 89}

\sqrt{4} \cdot \sqrt{89}

2 \cdot \sqrt{89}

ⅰ

2 \cdot 16

2 \cdot (10 + 6)

2 \cdot 10 + 2 \cdot 6

20 + 12

32

t_{1} = \frac{6 - 2 \cdot \sqrt{89}}{32}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{6 - 2 \cdot \sqrt{89}}{32}

\frac{3 - \sqrt{89}}{16}

t_{1} = \frac{3 - \sqrt{89}}{16}

The quadratic formula gives

t_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-6) + \sqrt{356}}{2 \cdot 16}

t_{2} = \frac{6 + \sqrt{356}}{2 \cdot 16}

ⓘ We look for the largest square dividing

356

4

2^{2}

, and take it out of the root:

\sqrt{356}

\sqrt{4 \cdot 89}

\sqrt{4} \cdot \sqrt{89}

2 \cdot \sqrt{89}

ⅰ

2 \cdot 16

2 \cdot (10 + 6)

2 \cdot 10 + 2 \cdot 6

20 + 12

32

t_{2} = \frac{6 + 2 \cdot \sqrt{89}}{32}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{6 + 2 \cdot \sqrt{89}}{32}

\frac{3 + \sqrt{89}}{16}

t_{2} = \frac{3 + \sqrt{89}}{16}

▶Solutions obtained:

t = {\frac{3 - \sqrt{89}}{16}, \frac{3 + \sqrt{89}}{16}}

2) The numeric value of:

t = {\frac{3 - \sqrt{89}}{16}, \frac{3 + \sqrt{89}}{16}}

is:

t = {\approx \frac{3 - 9.4340}{16}, \approx \frac{3 + 9.4340}{16}}

t = {\approx - \frac{6.4340}{16}, \approx \frac{12.4340}{16}}

▷

t = {\approx -0.4021, \approx 0.7771}