factor 4x^2+19x+21 — step-by-step solution

⚠Ⓘ 4x is read as the product 4*x: for a power, write 4^x‖19x is read as the product 19*x: for a power, write 19^x

Let's factor in

ℝ

4 x^{2} + 19 x + 21

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

Δ = 19^{2} - 4 \cdot 4 \cdot 21

ℹ

19^{2}

361

‖

4 \cdot 4

16

Δ = 361 - 16 \cdot 21

Δ = 361 - 336

Δ = 25

Since

Δ > 0

, the equation has two solutions:

The first solution is:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{- 19 - \sqrt{25}}{2 \cdot 4}

ℹ

- \sqrt{25}

-5

‖

2 \cdot 4

8

x1 = \frac{- 19 - 5}{8}

x1 = - \frac{24}{8}

x1 = -3

For the second solution, we obtain:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{- 19 + \sqrt{25}}{2 \cdot 4}

ℹ

\sqrt{25}

5

‖

2 \cdot 4

8

x2 = \frac{- 19 + 5}{8}

x2 = - \frac{14}{8}

x2 = - \frac{7}{4}

▶Factors identified:

4 \cdot (x + 3) \cdot (x + \frac{7}{4})