Here is Noah's work: 4x^2-20x+88=0 4x^2-20x=-88 4x^2-20x+25=-88+25 (2x-5)^2=-63 Show how Noah can finish his work using complex numbers.

Question

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Here is Noah's work: 4x^2-20x+88=0 4x^2-20x=-88 4x^2-20x+25=-88+25 (2x-5)^2=-63 Show how Noah can finish his work using complex numbers.

Answer 1

The roots of \((2x-5)^2 = -63\) are given by: \(x = (5 + \sqrt{-63})/2\) (complex root) and \(x = (5 - \sqrt{-63})/2\) (complex root).

Answer 2

## Step 1 Noah started with the equation \(4x^2 - 20x + 88 = 0\). After factoring, he obtained \((2x-5)^2 = -63\). ## Step 2 To solve this equation for \(x\), we need to take the square root of both sides. Since the square root of a negative number is not a real number, we introduce complex numbers. ### **Complex numbers are in the form \(a + bi\), where \(a\) is the "real" part of the number and \(b\) is the "imaginary" part of the number.** ## Step 3 We can rewrite \(\sqrt{-63}\) as \(0 + \sqrt{63}i = 0 + 7.93725i\). ## Step 4 Now, we solve for \(x\) by adding 5 to both sides and then dividing by 2. This gives us two solutions: ### **\(x_1 = (5 + \sqrt{-63})/2\)** and ### **\(x_2 = (5 - \sqrt{-63})/2\)** These are the solutions to the equation, and they are complex roots.

Question

Here is Noah's work: 4x^2-20x+88=0 4x^2-20x=-88 4x^2-20x+25=-88+25 (2x-5)^2=-63 Show how Noah can finish his work using complex numbers.

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