Substitute the values for *a*, *b*, and *c* into *b*2 – 4*ac* to determine the discriminant. Which quadratic equations will have two real number solutions? (The related quadratic function will have two *x*-intercepts.) Check all that apply. 

0 = 2*x*2 – 7*x* – 9

0 = *x*2 – 4*x* + 4

0 = 4*x*2 – 3*x* – 1

0 = *x*2 – 2*x* – 8

0 = 3*x*2 + 5*x* + 3

To determine which quadratic equations will have two real solutions, we need to calculate the discriminant (\(\Delta\)) for each equation. The discriminant formula is:

\[
\Delta = b^2 - 4ac
\]

- If \(\Delta > 0\), the quadratic equation has **two real and distinct solutions** (or two \(x\)-intercepts).
- If \(\Delta = 0\), the quadratic equation has **one real solution** (or one \(x\)-intercept).
- If \(\Delta < 0\), the quadratic equation has **no real solutions** (no \(x\)-intercepts).

Now, let's calculate the discriminant for each equation.

---

### 1. \( 0 = 2x^2 - 7x - 9 \)

Here, \(a = 2\), \(b = -7\), and \(c = -9\).

\[
\Delta = (-7)^2 - 4(2)(-9) = 49 + 72 = 121
\]

Since \(\Delta = 121 > 0\), this equation has **two real solutions**.

---

### 2. \( 0 = x^2 - 4x + 4 \)

Here, \(a = 1\), \(b = -4\), and \(c = 4\).

\[
\Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0
\]

Since \(\Delta = 0\), this equation has **one real solution**.

---

### 3. \( 0 = 4x^2 - 3x - 1 \)

Here, \(a = 4\), \(b = -3\), and \(c = -1\).

\[
\Delta = (-3)^2 - 4(4)(-1) = 9 + 16 = 25
\]

Since \(\Delta = 25 > 0\), this equation has **two real solutions**.

---

### 4. \( 0 = x^2 - 2x - 8 \)

Here, \(a = 1\), \(b = -2\), and \(c = -8\).

\[
\Delta = (-2)^2 - 4(1)(-8) = 4 + 32 = 36
\]

Since \(\Delta = 36 > 0\), this equation has **two real solutions**.

---

### 5. \( 0 = 3x^2 + 5x + 3 \)

Here, \(a = 3\), \(b = 5\), and \(c = 3\).

\[
\Delta = (5)^2 - 4(3)(3) = 25 - 36 = -11
\]

Since \(\Delta = -11 < 0\), this equation has **no real solutions**.

---

### Summary:

The quadratic equations with **two real solutions** are:

- \( 0 = 2x^2 - 7x - 9 \)
- \( 0 = 4x^2 - 3x - 1 \)
- \( 0 = x^2 - 2x - 8 \)

Would you like more details or clarifications?

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Here are 5 related questions:
1. What happens if the discriminant is negative in a quadratic equation?
2. How can the quadratic formula be used to find real solutions?
3. How does the graph of a quadratic equation relate to its discriminant?
4. Can a quadratic equation with a discriminant of zero have complex solutions?
5. What is the relationship between the coefficients and the shape of the parabola?

**Tip:** The discriminant not only tells the number of solutions but also helps determine the nature of the solutions—whether they are real or complex.

Substitute the values for a, b, and c into b^2 – 4ac to determine the discriminant. Which quadratic equations will have two real number solutions? (The related quadratic function will have two x-intercepts.) Check all that apply.

0 = 2x^2 – 7x – 9

0 = x^2 – 4x + 4

0 = 4x^2 – 3x – 1

0 = x^2 – 2x – 8

0 = 3x^2 + 5x + 3

Learn how to determine which quadratic equations have two real solutions by using the discriminant formula Δ = b^2 - 4ac. This detailed explanation includes examples of equations like 2x^2 – 7x – 9 and x^2 – 2x – 8, showing step-by-step how to calculate the discriminant and interpret its value. Suitable for students in Grades 8-10.

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