intercepts of f(x)=-x^2+5x+6 — step-by-step solution

⚠Ⓘ 5x is read as the product 5*x: for a power, write 5^x

1) On the horizontal axis, y = 0: solve

- x^{2} + 5 x + 6 = 0

⇥1.1) Let's solve in

ℝ

- x^{2} + 5 x + 6 = 0

⇥Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

⇥

Δ = 5^{2} - 4 \cdot (-1) \cdot 6

⇥ℹ

5^{2}

25

‖

-4 \cdot (-1)

4

⇥

Δ = 25 + 4 \cdot 6

⇥

Δ = 25 + 24

⇥

Δ = 49

⇥As

Δ > 0

, the equation has two solutions:

⇥For the first solution, we obtain:

\frac{- b - \sqrt{Δ}}{2 a}

⇥

x1 = \frac{- 5 - \sqrt{49}}{2 \cdot (-1)}

⇥ℹ

- \sqrt{49}

-7

‖

2 \cdot (-1)

-2

⇥

x1 = \frac{- 5 - 7}{-2}

⇥

x1 = - (\frac{- 5 - 7}{2})

⇥

x1 = - (\frac{-12}{2})

⇥

x1 = \frac{12}{2}

⇥

x1 = 6

⇥For the second solution, we obtain:

\frac{- b + \sqrt{Δ}}{2 a}

⇥

x2 = \frac{- 5 + \sqrt{49}}{2 \cdot (-1)}

⇥ℹ

\sqrt{49}

7

‖

2 \cdot (-1)

-2

⇥

x2 = \frac{- 5 + 7}{-2}

⇥

x2 = - (\frac{- 5 + 7}{2})

⇥

x2 = - \frac{2}{2}

⇥

x2 = -1

⇥Solution obtained:

x = -1

⇥Solution obtained:

x = 6

On the vertical axis, x = 0: calculate f(0)

⇥

- 0^{2} + 5 \cdot 0 + 6

⇥

0 + 6

⇥

6

▶Axis intercepts:

{(-1, 0), (6, 0), (0, 6)}