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Magazine Chapter 5: Problem 21
Solve each equation. $$ 3 m^{2}+10 m+3=0 $$
Short Answer
Expert verified
The solutions are \(m = -\frac{1}{3}\) and \(m = -3\).
Step by step solution
01
Identify the Quadratic Equation Format
The given equation is a quadratic equation in the standard form ax虏 + bx + c = 0. In this case, a = 3, b = 10, and c = 3.
02
Apply the Quadratic Formula
The quadratic formula, \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is a method we can use to solve for m. Substitute the values for a, b, and c into the formula: \(m = \frac{-10 \pm \sqrt{10^2 - 4(3)(3)}}{2(3)}\).
03
Calculate the Discriminant
The discriminant, \(b^2 - 4ac\), determines the nature of the roots. Calculate it: \(10^2 - 4(3)(3) = 100 - 36 = 64\). Since the discriminant is positive, there are two distinct real roots.
04
Solve for the Values of m
Substitute the discriminant back into the quadratic formula and solve: \(m = \frac{-10 \pm \sqrt{64}}{6}\). This gives \(m = \frac{-10 \pm 8}{6}\).
05
Calculate the Two Roots Separately
First, calculate \(m = \frac{-10 + 8}{6}\) which is \(m = \frac{-2}{6} = -\frac{1}{3}\). Then, calculate \(m = \frac{-10 - 8}{6}\) which is \(m = \frac{-18}{6} = -3\).
06
Verify the Solutions
Substitute \(m = -\frac{1}{3}\) and \(m = -3\) back into the original equation to verify that both are solutions. They satisfy the equation, confirming they are correct.
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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 solving quadratic equations using the quadratic formula. It is represented as the expression \( b^2 - 4ac \). This component aids in predicting the nature of the roots of the quadratic equation even before solving it.
- If the discriminant is positive, you will find two distinct real roots.
- If the discriminant is zero, the quadratic equation will have exactly one real root, also known as a double root.
- On the other hand, if the discriminant is negative, there will be no real roots, and the roots will instead be complex or imaginary numbers.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations when factoring is difficult or impossible. It is given by:\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula calculates the roots of any quadratic equation in the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are the coefficients from the equation. By merely substituting these values into the formula, you can directly calculate the roots with ease.
In the given problem, we have values:\ - \(a = 3\)- \(b = 10\), and - \(c = 3\).By plugging them into the quadratic formula, it streamlines the process and eliminates the need for trial and error methods.
In the given problem, we have values:\ - \(a = 3\)- \(b = 10\), and - \(c = 3\).By plugging them into the quadratic formula, it streamlines the process and eliminates the need for trial and error methods.
Roots of Quadratic Equation
The roots of a quadratic equation are the solutions that satisfy the equation \( ax^2 + bx + c = 0 \). These roots can be real or complex, depending on the discriminant as discussed earlier.
In the given exercise, substituting back into the quadratic formula \( m = \frac{-10 \pm \sqrt{64}}{6} \) gives you two values for \( m \):
In the given exercise, substituting back into the quadratic formula \( m = \frac{-10 \pm \sqrt{64}}{6} \) gives you two values for \( m \):
- \( m = -\frac{1}{3} \) and
- \( m = -3 \).
