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Chapter 1: Problem 66

Find all real solutions of the quadratic equation. $$x^{2}+30 x+200=0$$

Short Answer

Expert verified
The real solutions are \(x = -10\) and \(x = -20\).

Step by step solution

01

Identify coefficients

In the quadratic equation given by \(x^2 + 30x + 200 = 0\), identify the coefficients. Here, \(a = 1\), \(b = 30\), and \(c = 200\).
02

Calculate the discriminant

The discriminant \(\Delta\) is calculated as \(b^2 - 4ac\). Substitute the values: \(\Delta = 30^2 - 4 \times 1 \times 200 = 900 - 800 = 100\).
03

Determine the nature of the roots

Since the discriminant \(\Delta = 100\) is greater than zero, it means the quadratic equation has two distinct real roots.
04

Apply the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\). Substitute \(b = 30\), \(\Delta = 100\), and \(a = 1\) into the formula: \(x = \frac{-30 \pm \sqrt{100}}{2}\).
05

Simplify to find the solutions

Solve for \(x\):\(x = \frac{-30 + 10}{2} = \frac{-20}{2} = -10\) and \(x = \frac{-30 - 10}{2} = \frac{-40}{2} = -20\).Therefore, the solutions are \(x = -10\) and \(x = -20\).

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Key Concepts

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

Discriminant
In any quadratic equation, the discriminant is a crucial component. It helps to determine the nature of the roots. The discriminant is denoted by \( \Delta \), and in a quadratic equation like \( ax^2 + bx + c = 0 \), it is calculated as \( b^2 - 4ac \). This simple formula tells us a lot about the equation's roots.
  • If \( \Delta > 0 \), there are two distinct real roots. This is because the square root of a positive number is real, providing two different solutions when added and subtracted.
  • If \( \Delta = 0 \), there is one real root, also known as a repeated or double root. This happens when the parabola touches the x-axis at one point.
  • If \( \Delta < 0 \), there are no real roots, only complex or imaginary roots, since the square root of a negative number involves imaginary numbers.
For the given equation \( x^2 + 30x + 200 = 0 \), the discriminant is \( 100 \), indicating two distinct real roots.
Quadratic Formula
The quadratic formula is a tool we use to find solutions of any quadratic equation. It allows us to compute the roots directly, using the equation's coefficients without needing to factorize. The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here鈥檚 how it applies to our example:
  • Given \( a = 1 \), \( b = 30 \), and \( c = 200 \), the discriminant \( \Delta \) is calculated as \( 100 \).
  • We plug these into the formula: \( x = \frac{-30 \pm \sqrt{100}}{2} \).
  • This leads to two potential solutions due to the \( \pm \) (plus-minus) operator, which accounts for both roots.
Using the quadratic formula is a systematic way to find all possible roots of quadratics, especially when factoring is challenging or impossible.
Roots of Quadratics
Finding the roots of quadratic equations means solving them for their \( x \)-values where the equation equals zero. These roots are essentially the x-intercepts where the graph of the equation touches or crosses the x-axis. For most quadratics, there are two roots, but they can be real, repeated, or complex based on the discriminant.
The specific process is as follows:
  • Calculate the discriminant \( \Delta \).
  • Use the quadratic formula \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \) to find the roots.
  • In our example, we find \( x = \frac{-30 + 10}{2} = -10 \) and \( x = \frac{-30 - 10}{2} = -20 \).
These values, \( x = -10 \) and \( x = -20 \), indicate the points where the graph of the quadratic equation \( x^2 + 30x + 200 = 0 \) intersects the x-axis. Understanding how to derive these roots underlines your ability to solve quadratic equations more generally.

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