4x^2+16x-9=0 — step-by-step solution

1) Let's solve in

ℝ

4 x^{2} + 16 x - 9 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 4

b = 16

and

c = -9

Δ = 16^{2} - 4 \cdot 4 \cdot (-9)

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

16^{2}

16 \cdot 16

256

■Multiplications are done before additions and subtractions:

4 \cdot 4

16

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

- 16 \cdot (-9)

16 \cdot 9

Δ = 256 + 16 \cdot 9

ⓘ Multiplications are done before additions and subtractions:

16 \cdot 9

(10 + 6) \cdot 9

10 \cdot 9 + 6 \cdot 9

90 + 54

144

Δ = 256 + 144

ⓘ Start from

256

and move forward

144

on the number line:

256 + 144

400

Δ = 400

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 16 - \sqrt{400}}{2 \cdot 4}

x_{1} = \frac{- 16 - \sqrt{400}}{2 \cdot 4}

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

\sqrt{400}

20

;

20^{2}

400

ⅰ

2 \cdot 4

8

x_{1} = \frac{- 16 - 20}{8}

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

- 16 - 20

-36

x_{1} = - \frac{36}{8}

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

4

, their greatest common divisor:

- \frac{36}{8}

- \frac{9 \cdot 4}{2 \cdot 4}

- \frac{9}{2}

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 16 + \sqrt{400}}{2 \cdot 4}

x_{2} = \frac{- 16 + \sqrt{400}}{2 \cdot 4}

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

\sqrt{400}

20

;

20^{2}

400

ⅰ

2 \cdot 4

8

x_{2} = \frac{- 16 + 20}{8}

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

- 16 + 20

4

x_{2} = \frac{4}{8}

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

4

, their greatest common divisor:

\frac{4}{8}

\frac{1 \cdot 4}{2 \cdot 4}

\frac{1}{2}

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

▶Solutions obtained:

x = {- \frac{9}{2}, \frac{1}{2}}

2) The numeric value of:

x = {- \frac{9}{2}, \frac{1}{2}}

equals:

▷

x = {-4.5, 0.5}