Find the zeroes of quadratic polynomial 4√5x²-17x+3√5
and verify the relationship 
between the zeroes and the coefficients. 

To find the zeroes of the quadratic polynomial \( 4\sqrt{5}x^2 - 17x + 3\sqrt{5} \), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 4\sqrt{5} \), \( b = -17 \), and \( c = 3\sqrt{5} \).

### Step 1: Identify coefficients
\[
a = 4\sqrt{5}, \quad b = -17, \quad c = 3\sqrt{5}
\]

### Step 2: Calculate the discriminant
\[
\Delta = b^2 - 4ac
\]
Substitute the values of \( a \), \( b \), and \( c \):

\[
\Delta = (-17)^2 - 4(4\sqrt{5})(3\sqrt{5})
\]
\[
\Delta = 289 - 4(4)(3)(5)
\]
\[
\Delta = 289 - 240
\]
\[
\Delta = 49
\]

### Step 3: Apply the quadratic formula
Now, use the quadratic formula to find the zeroes:

\[
x = \frac{-(-17) \pm \sqrt{49}}{2(4\sqrt{5})}
\]
\[
x = \frac{17 \pm 7}{8\sqrt{5}}
\]

#### Zero 1:
\[
x_1 = \frac{17 + 7}{8\sqrt{5}} = \frac{24}{8\sqrt{5}} = \frac{3}{\sqrt{5}} = \frac{3\sqrt{5}}{5}
\]

#### Zero 2:
\[
x_2 = \frac{17 - 7}{8\sqrt{5}} = \frac{10}{8\sqrt{5}} = \frac{5}{4\sqrt{5}} = \frac{5\sqrt{5}}{20} = \frac{\sqrt{5}}{4}
\]

Thus, the zeroes of the quadratic polynomial are \( x_1 = \frac{3\sqrt{5}}{5} \) and \( x_2 = \frac{\sqrt{5}}{4} \).

### Step 4: Verify the relationship between the zeroes and the coefficients
For any quadratic polynomial \( ax^2 + bx + c \), the sum and product of the zeroes are related to the coefficients as follows:

1. **Sum of the zeroes**:
\[
x_1 + x_2 = -\frac{b}{a}
\]
2. **Product of the zeroes**:
\[
x_1 \cdot x_2 = \frac{c}{a}
\]

#### Sum of the zeroes:
\[
x_1 + x_2 = \frac{3\sqrt{5}}{5} + \frac{\sqrt{5}}{4}
\]
To add these, find the common denominator:
\[
x_1 + x_2 = \frac{12\sqrt{5}}{20} + \frac{5\sqrt{5}}{20} = \frac{17\sqrt{5}}{20}
\]
Now calculate \( -\frac{b}{a} \):
\[
-\frac{b}{a} = -\frac{-17}{4\sqrt{5}} = \frac{17}{4\sqrt{5}} = \frac{17\sqrt{5}}{20}
\]
The sum of the zeroes matches \( \frac{17\sqrt{5}}{20} \), as expected.

#### Product of the zeroes:
\[
x_1 \cdot x_2 = \frac{3\sqrt{5}}{5} \cdot \frac{\sqrt{5}}{4} = \frac{3 \cdot (\sqrt{5})^2}{5 \cdot 4} = \frac{3 \cdot 5}{20} = \frac{15}{20} = \frac{3}{4}
\]
Now calculate \( \frac{c}{a} \):
\[
\frac{c}{a} = \frac{3\sqrt{5}}{4\sqrt{5}} =

Find the zeroes of quadratic polynomial 4√5x²-17x+3√5 and verify the relationship between the zeroes and the coefficients.

Learn how to find the zeroes of the quadratic polynomial 4√5x² - 17x + 3√5 and verify the relationship between the zeroes and coefficients using the quadratic formula. This detailed solution includes key algebraic concepts and is suitable for students in Grades 9-12.

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