100x^2-160x+63=0 — step-by-step solution

1) Let's solve in

ℝ

100 x^{2} - 160 x + 63 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 100

b = -160

and

c = 63

Δ = {(-160)}^{2} - 4 \cdot 100 \cdot 63

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

{(-160)}^{2}

(-160) \cdot (-160)

25,600

■Multiplications are done before additions and subtractions:

4 \cdot 100

400

Δ = 25,600 - 400 \cdot 63

ⓘ Multiplications are done before additions and subtractions:

400 \cdot 63

400 \cdot (60 + 3)

400 \cdot 60 + 400 \cdot 3

24,000 + 1200

25,200

Δ = 25,600 - 25,200

ⓘ Start from

25,600

and move back

25,200

on the number line:

25,600 - 25,200

400

Δ = 400

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-160) - \sqrt{400}}{2 \cdot 100}

x_{1} = \frac{160 - \sqrt{400}}{2 \cdot 100}

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

\sqrt{400}

20

;

20^{2}

400

ⅰ

2 \cdot 100

200

x_{1} = \frac{160 - 20}{200}

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

160 - 20

140

x_{1} = \frac{140}{200}

We divide the numerator and denominator of

\frac{140}{200}

by the common factor:

20

, which gives the new quotient:

\frac{7}{10}

x_{1} = \frac{7}{10}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-160) + \sqrt{400}}{2 \cdot 100}

x_{2} = \frac{160 + \sqrt{400}}{2 \cdot 100}

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

\sqrt{400}

20

;

20^{2}

400

ⅰ

2 \cdot 100

200

x_{2} = \frac{160 + 20}{200}

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

160 + 20

180

x_{2} = \frac{180}{200}

The fraction:

\frac{180}{200}

simplifies by the common factor:

20

into the new fraction:

\frac{9}{10}

x_{2} = \frac{9}{10}

▶Solutions obtained:

x = {\frac{7}{10}, \frac{9}{10}}

2) The numeric value of:

x = {\frac{7}{10}, \frac{9}{10}}

equals:

▷

x = {0.7, 0.9}