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Chapter 6: Problem 2

A number is 30 less than its square. Find all such numbers.

Short Answer

Expert verified
The numbers are 6 and -5.

Step by step solution

01

Set up the Equation

Let the unknown number be denoted by the variable \(x\). According to the problem, the number is 30 less than its square. This can be translated into the equation: \[x = x^2 - 30\].
02

Rearrange the Equation

To solve the equation, rearrange it to standard quadratic form by moving all terms to one side: \[x^2 - x - 30 = 0\].
03

Solve the Quadratic Equation Using the Quadratic Formula

The quadratic formula is given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. For our equation \(x^2 - x - 30 = 0\), we have \(a = 1\), \(b = -1\), and \(c = -30\). Plug these values into the quadratic formula: \[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-30)}}{2(1)} = \frac{1 \pm \sqrt{1 + 120}}{2} = \frac{1 \pm \sqrt{121}}{2} \].
04

Simplify the Solutions

Simplify the expression under the square root and solve for \(x\): \[ x = \frac{1 \pm 11}{2} \]. This gives us two solutions: \[ x = \frac{1 + 11}{2} = 6 \] and \[ x = \frac{1 - 11}{2} = -5 \].
05

Verify the Solutions

To verify, substitute \(x = 6\) back into the original condition: \(6 = 6^2 - 30\) which simplifies to \(6 = 36 - 30 = 6\). Also, substitute \(x = -5\): \(-5 = (-5)^2 - 30\) which simplifies to \(-5 = 25 - 30 = -5\). Both solutions are correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Formula
To solve quadratic equations, the quadratic formula is a critical tool. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). Here, \

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