Question

Find the discriminant for each equation.
Explain what it tells you about the equation's roots and graph.
a) 5x'+4x-3=0
Discriminant = b2 - 4ac
Anower
Roots: The equation has
choose your answer...
real roots.
History.
Mastery.
Studio
Help
Graph: The graph crosses the x-axis in choose your answer...
places.
b) 9x2+6x+1=0
Discriminant = b2 - 4ac
1 Answer
Roots: The equation has
choose your answer... real roots.
Graph: The graph crosses the x-axis in choose your answer...
places.
c) x'+2x+7=0
Discriminant = b2 - 4ac
i Answer
-----
Roots: The equation has
Graph: The graph axis.
:: 36-4(9)(1) = 0
choose your answer..
choose your answer...
real roots.
touches/crosses the x-
:: 16-4(5)(-3) = 76
::4-4(1)(7) =-24

Answer 1

To find the discriminant for each equation, we'll use the formula:

\[
\text{Discriminant} = b^2 - 4ac
\]

where \( a \), \( b \), and \( c \) are the coefficients from the standard quadratic equation \( ax^2 + bx + c = 0 \).

### a) \( 5x^2 + 4x - 3 = 0 \)

For this equation, we identify:
- \( a = 5 \)
- \( b = 4 \)
- \( c = -3 \)

Calculating the discriminant:

\[
D = 4^2 - 4(5)(-3) = 16 + 60 = 76
\]

**Roots:** The equation has **two distinct real roots** (since the discriminant is positive).

**Graph:** The graph crosses the x-axis in **two places**.

### b) \( 9x^2 + 6x + 1 = 0 \)

For this equation, we have:
- \( a = 9 \)
- \( b = 6 \)
- \( c = 1 \)

Calculating the discriminant:

\[
D = 6^2 - 4(9)(1) = 36 - 36 = 0
\]

**Roots:** The equation has **one real root** (a double root) (since the discriminant is zero).

**Graph:** The graph crosses the x-axis in **one place** (it touches at that point).

### c) \( x^2 + 2x + 7 = 0 \)

For this equation, we identify:
- \( a = 1 \)
- \( b = 2 \)
- \( c = 7 \)

Calculating the discriminant:

\[
D = 2^2 - 4(1)(7) = 4 - 28 = -24
\]

**Roots:** The equation has **no real roots** (since the discriminant is negative).

**Graph:** The graph does not cross the x-axis (it stays above the x-axis).

### Summary:
- **a)** \( D = 76 \): Two distinct real roots; crosses x-axis in 2 places.
- **b)** \( D = 0 \): One real root; touches x-axis in 1 place.
- **c)** \( D = -24 \): No real roots; does not cross x-axis.