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Chapter 1: Problem 78

Find all real solutions of the quadratic equation. $$25 x^{2}+70 x+49=0$$

Short Answer

Expert verified
The real solution is \(x = -1.4\) (a repeated root).

Step by step solution

01

Recognize the Quadratic Form

Identify that the equation, \(25x^2 + 70x + 49 = 0\), is a standard quadratic equation in the form \(ax^2 + bx + c = 0\), where \(a = 25\), \(b = 70\), and \(c = 49\).
02

Calculate the Discriminant

Use the discriminant formula \(b^2 - 4ac\) to determine the nature of the roots. Calculate the discriminant: \(70^2 - 4 \times 25 \times 49 = 4900 - 4900 = 0\).
03

Analyze the Discriminant

Since the discriminant is 0, this means that the quadratic has exactly one real repeated root (also known as a double root).
04

Use the Quadratic Formula for the Root

Use the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\) to find the root. Since the discriminant is 0, the equation simplifies to \(x = \frac{{-b}}{2a}\).
05

Calculate the Solution

Substitute the values \(a = 25\) and \(b = 70\) into the simplified quadratic formula: \(x = \frac{{-70}}{2 \times 25}\). This simplifies to \(x = \frac{{-70}}{50} = -1.4\).

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Key Concepts

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

Discriminant
The discriminant is a crucial part of understanding the nature of the solutions, or "roots," of a quadratic equation. In a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is given by the formula \( b^2 - 4ac \). Calculating the discriminant helps determine how many and what type of solutions exist for the quadratic equation.
  • If \( D > 0 \), there are two distinct real roots.
  • If \( D = 0 \), there is exactly one real root, which is a repeated root.
  • If \( D < 0 \), there are no real roots, but two complex roots.
In the given exercise, the discriminant is calculated as \( 70^2 - 4 \times 25 \times 49 = 0 \). This tells us that the quadratic equation has a double root, which means it touches the x-axis at exactly one point.
Root Types
Understanding root types is essential for solving quadratic equations. The roots of a quadratic equation can tell us much about the graph of the quadratic function. Based on the value of the discriminant, quadratic equations can have different root types:
  • Distinct Real Roots: Occurs when the discriminant \( D \) is greater than zero, resulting in two separate points where the graph crosses the x-axis.
  • Repeated Root: Occurs when \( D = 0 \). This type occurs once, and the graph just touches the x-axis, implying the vertex lies on the x-axis.
  • Complex Roots: Occurs when \( D < 0 \). In this case, the graph does not intersect the x-axis, as the roots are complex, involving imaginary numbers.
Since the discriminant of the exercise is zero, we deal with a repeated root, yielding a vertex that just touches the x-axis, specifically at the value \( x = -1.4 \).
Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). It is defined as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula works regardless of whether the roots are real or complex. However, the discriminant values alter the calculation slightly.
When using the quadratic formula:
  • If \( D > 0 \), you use the \( \pm \) to find two different solutions.
  • If \( D = 0 \), the formula simplifies to \( x = \frac{-b}{2a} \), indicating a repeated root.
  • If \( D < 0 \), the square root of a negative number leads to complex roots.
In the equation \( 25x^2 + 70x + 49 = 0 \), with a zero discriminant, the quadratic formula simplifies straightaway to \( x = \frac{-b}{2a} \). Substituting \( a = 25 \) and \( b = 70 \), we found the root to be \( x = -1.4 \), confirming its status as the unique real solution.

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