Show that the following pair of equations has a common root .b.)3x2-8x+4=0,4x2-7x-2=0

To show that the pair of equations has a common root, we will find the roots of both equations and see if they share at least one root.

Given equations:
1. \( 3x^2 - 8x + 4 = 0 \)
2. \( 4x^2 - 7x - 2 = 0 \)

### Step 1: Find the roots of the first equation
The first equation is:
\[
3x^2 - 8x + 4 = 0
\]

Use the quadratic formula to find the roots:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 3 \), \( b = -8 \), and \( c = 4 \).

Substituting these values into the quadratic formula:
\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(3)(4)}}{2(3)}
\]
\[
x = \frac{8 \pm \sqrt{64 - 48}}{6}
\]
\[
x = \frac{8 \pm \sqrt{16}}{6}
\]
\[
x = \frac{8 \pm 4}{6}
\]

This gives us two roots:
\[
x_1 = \frac{8 + 4}{6} = \frac{12}{6} = 2
\]
\[
x_2 = \frac{8 - 4}{6} = \frac{4}{6} = \frac{2}{3}
\]

### Step 2: Find the roots of the second equation
The second equation is:
\[
4x^2 - 7x - 2 = 0
\]

Again, use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 4 \), \( b = -7 \), and \( c = -2 \).

Substituting these values into the quadratic formula:
\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(4)(-2)}}{2(4)}
\]
\[
x = \frac{7 \pm \sqrt{49 + 32}}{8}
\]
\[
x = \frac{7 \pm \sqrt{81}}{8}
\]
\[
x = \frac{7 \pm 9}{8}
\]

This gives us two roots:
\[
x_1 = \frac{7 + 9}{8} = \frac{16}{8} = 2
\]
\[
x_2 = \frac{7 - 9}{8} = \frac{-2}{8} = -\frac{1}{4}
\]

### Step 3: Compare the roots
The roots of the first equation are \( 2 \) and \( \frac{2}{3} \).
The roots of the second equation are \( 2 \) and \( -\frac{1}{4} \).

Since both equations share the root \( x = 2 \), we have shown that these two quadratic equations have a common root.

### 5 Related Questions:
1. How do you determine if two quadratic equations have common roots using their discriminants?
2. Can two different quadratic equations have more than one common root?
3. What is the geometric interpretation of having a common root for two quadratic equations?
4. How can the Vieta's formulas be used to find the sum and product of the roots of a quadratic equation?
5. What methods can be used to solve a system of nonlinear equations involving quadratics?

### Tip:
When checking for common roots between two quadratic equations, using the quadratic formula or factoring can quickly reveal any shared roots.

Learn how to demonstrate that the quadratic equations 3x^2 - 8x + 4 = 0 and 4x^2 - 7x - 2 = 0 have a common root. Step-by-step solution using the quadratic formula provided. Suitable for high school students and beyond.

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