Question

identify all of the root(s) of \\(g(x) = (x^2 + 3x - 4)(x^2 - 4x + 29)\\).

-1
1
-4
4
2 + 5i
2 - 5i
-2 + 10i
-2 - 10i

Explanation:

🆕 New Concept Discovered: Complex Roots of Quadratic Equations
Finding roots when the discriminant is negative.

Step 1: Set each factor to zero

To find the roots of \( g(x) = (x^2 + 3x - 4)(x^2 - 4x + 29) \), we set the function equal to zero:

$$ (x^2 + 3x - 4)(x^2 - 4x + 29) = 0 $$

This gives us two separate quadratic equations to solve:

1. \( x^2 + 3x - 4 = 0 \)
2. \( x^2 - 4x + 29 = 0 \)

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Step 2: Solve the first quadratic equation

We solve \( x^2 + 3x - 4 = 0 \) by factoring. We look for two numbers that multiply to \(-4\) and add to \(3\). These numbers are \(4\) and \(-1\):

$$ (x + 4)(x - 1) = 0 $$

Setting each factor to zero gives:

$$ x = -4 $$

$$ x = 1 $$

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Step 3: Solve the second quadratic equation

We solve \( x^2 - 4x + 29 = 0 \) using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -4 \), and \( c = 29 \):

$$ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(29)}}{2(1)} $$

$$ x = \frac{4 \pm \sqrt{16 - 116}}{2} $$

$$ x = \frac{4 \pm \sqrt{-100}}{2} $$

Since we have a negative number under the square root, we use the imaginary unit \( i \) (where \( \sqrt{-1} = i \)):

$$ \sqrt{-100} = 10i $$

Substitute this back into the equation:

$$ x = \frac{4 \pm 10i}{2} $$

$$ x = 2 \pm 5i $$

This gives two complex roots:

$$ x = 2 + 5i $$

$$ x = 2 - 5i $$

Answer:

The roots of the function are:

• \(-4\)
• \(1\)
• \(2 + 5i\)
• \(2 - 5i\)