Is the valre of the discimimant, $$ b^{2}-4 a c $$, for the quadratic equation $$ 0=x^{2}-4 x+5 $$, and what does umber of real solutions the equation has? Tho disctiminant is -4 , so the equation has 2 real solutions, The disctiminant is $$ =4 $$, so the eqpation has no real solutions, The disctiminant is 35 , so the equation has 2 real solutions. The discrininant is 35 so the equalion las no real solutions.

Question

UpStudy AI · Accepted Answer

To determine the discriminant of the quadratic equation and the number of real solutions it has, let's follow the steps below.

### **Given Quadratic Equation:**
$$ 0 = x^{2} - 4x + 5 $$

### **Step 1: Identify the Coefficients**
A general quadratic equation is of the form:
$$ ax^{2} + bx + c = 0 $$
Comparing this with your equation:
- $$ a = 1 $$
- $$ b = -4 $$
- $$ c = 5 $$

### **Step 2: Calculate the Discriminant**
The discriminant ($$ D $$) is calculated using the formula:
$$ D = b^{2} - 4ac $$
Plugging in the values:
$$ D = (-4)^{2} - 4(1)(5) $$
$$ D = 16 - 20 $$
$$ D = -4 $$

### **Step 3: Determine the Number of Real Solutions**
The discriminant tells us about the nature of the roots of the quadratic equation:
- **If $$ D > 0 $$:** There are **two distinct real solutions**.
- **If $$ D = 0 $$:** There is **exactly one real solution** (a repeated root).
- **If $$ D < 0 $$:** There are **no real solutions**; instead, there are two complex (imaginary) solutions.

In this case, since $$ D = -4 $$ (which is less than 0), the equation has **no real solutions**.

### **Summary:**
- **Discriminant ($$ D $$) = -4**
- **Number of Real Solutions:** 0 (No real solutions; two complex solutions exist)

### **Clarifying the Options Provided:**
Let's review the options you've mentioned:

1. **The discriminant is -4, so the equation has 2 real solutions.**
   - **Incorrect.** A negative discriminant means **no real solutions**.

2. **The discriminant is 4, so the equation has no real solutions.**
   - **Incorrect.** If $$ D = 4 $$ (positive), there **are two real solutions**.

3. **The discriminant is 35, so the equation has 2 real solutions.**
   - **Partially Incorrect.** While a positive discriminant does indicate two real solutions, the actual discriminant is **-4**, not 35.

4. **The discriminant is 35, so the equation has no real solutions.**
   - **Incorrect.** A positive discriminant like 35 would mean **two real solutions**, not none.

### **Correct Statement:**
**The discriminant is -4, so the equation has no real solutions.**
The discriminant is -4, so the equation has no real solutions.

Upstudy AI Solution

Answer

Solution

Given Quadratic Equation:

Step 1: Identify the Coefficients

Step 2: Calculate the Discriminant

Step 3: Determine the Number of Real Solutions

Summary:

Clarifying the Options Provided:

Correct Statement:

Extra Insights

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