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Magazine Chapter 6: Problem 56
Solve each quadratic equation using the method that seems most appropriate. $$ 9 x^{2}+18 x+5=0 $$
Short Answer
Expert verified
The roots are \(x = -\frac{1}{3}\) and \(x = -\frac{5}{3}\).
Step by step solution
01
Identify the Standard Form
The given quadratic equation is in the form \(ax^2 + bx + c = 0\), where \(a = 9\), \(b = 18\), and \(c = 5\).
02
Calculate the Discriminant
The discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). Plugging in the values, we have \(D = 18^2 - 4(9)(5) = 324 - 180 = 144\).
03
Check the Discriminant
Since the discriminant \(D = 144\) is a perfect square, it is suitable to use the quadratic formula for finding the real roots.
04
Apply the Quadratic Formula
The quadratic formula is \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substituting the values, we have \(x = \frac{-18 \pm \sqrt{144}}{18}\).
05
Simplify the Expression
First, calculate \(\sqrt{144} = 12\). Now substitute this back into the quadratic formula to get \(x = \frac{-18 \pm 12}{18}\). This will give two solutions: \(x = \frac{-18 + 12}{18}\) and \(x = \frac{-18 - 12}{18}\).
06
Solve for Both Roots
For the first root, calculate \(x = \frac{-6}{18} = -\frac{1}{3}\). For the second root, calculate \(x = \frac{-30}{18} = -\frac{5}{3}\).
07
State the Final Roots
The solutions to the quadratic equation \(9x^2 + 18x + 5 = 0\) are \(x = -\frac{1}{3}\) and \(x = -\frac{5}{3}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula provides a direct method to find the roots, or solutions, of the equation without needing to factorize or complete the square.
The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
- \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.
- The term under the square root, \(b^2 - 4ac\), is known as the discriminant.
- The "\(\pm\)" symbol indicates that the quadratic formula provides two potential solutions for \(x\).
Discriminant
The discriminant is a key part of understanding the nature of the roots of a quadratic equation. It is the expression found under the square root in the quadratic formula:\[D = b^2 - 4ac\]The value of the discriminant determines the number and type of roots of the quadratic equation:
- If \(D > 0\), there are two distinct real roots. This means the quadratic equation will intersect the x-axis at two points.
- If \(D = 0\), there is exactly one real root, often referred to as a "repeated" or "double" root. The graph of the equation touches the x-axis at a point.
- If \(D < 0\), there are no real roots, which implies the quadratic equation does not intersect the x-axis, but instead, has two complex roots.
Real Roots
Real roots are solutions to a quadratic equation where the discriminant is positive or zero. These roots are the x-values where the graph of the quadratic equation intersects the x-axis.
These values are the points where the parabola represented by the equation cuts through the x-axis, illustrating two distinct real roots.
- When the discriminant \(D > 0\), the quadratic equation has two distinct real roots.
- When \(D = 0\), the equation has one real root, and the parabola touches the x-axis at a single point.
These values are the points where the parabola represented by the equation cuts through the x-axis, illustrating two distinct real roots.
