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Magazine Chapter 6: Problem 302
Solve. $$ 3 y^{2}-18 y=-27 $$
Short Answer
Expert verified
The solution is y = 3.
Step by step solution
01
Write the equation in standard form
First, write the given quadratic equation in standard form. The standard form of a quadratic equation is \[ ax^{2} + bx + c = 0 \]. Given the equation \[ 3y^{2} - 18y = -27 \], add 27 to both sides to get: \[ 3y^{2} - 18y + 27 = 0 \].
02
Identify coefficients
Identify the coefficients in the standard form equation \[ 3y^{2} - 18y + 27 = 0 \]. Here, \[ a = 3 \], \[ b = -18 \], and \[ c = 27 \].
03
Use the quadratic formula
Use the quadratic formula \[ y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] to find the solutions. Plug in the identified coefficients: \[ a = 3 \], \[ b = -18 \], and \[ c = 27 \].
04
Calculate the discriminant
Calculate the discriminant \[ b^{2} - 4ac \]. Here, \[ b^{2} = (-18)^{2} = 324 \] and \[ 4ac = 4 \cdot 3 \cdot 27 = 324 \]. Therefore, the discriminant is \[ 324 - 324 = 0 \].
05
Solve using the quadratic formula
Since the discriminant is 0, there is one real solution. Substitute into the quadratic formula to get: \[ y = \frac{-(-18) \pm \sqrt{0}}{2 \cdot 3} = \frac{18 \pm 0}{6} = 3 \]. So, the solution is \[ y = 3 \].
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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 used to find the solutions of a quadratic equation. A quadratic equation is typically in the form \[ ax^2 + bx + c = 0 \]. The quadratic formula itself is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula allows you to solve any quadratic equation by plugging in the coefficients a, b, and c. Let's break it down:
- 'a' is the coefficient of \( x^2 \)
- 'b' is the coefficient of \( x \)
- 'c' is the constant term
- Calculate the discriminant, \( b^2 - 4ac \)
- Substitute the values of a, b, and the discriminant into the formula
- Solve for x
discriminant
The discriminant is a key part of the quadratic formula. It determines the nature and number of the roots of a quadratic equation. The discriminant is the expression located under the square root in the quadratic formula, given by: \[ b^2 - 4ac \]. Here's what it tells us:
- If \( b^2 - 4ac > 0 \), there are two real and distinct solutions
- If \( b^2 - 4ac = 0 \), there is one real solution (or a repeated root)
- If \( b^2 - 4ac < 0 \), there are no real solutions, but two complex solutions
standard form of a quadratic equation
The standard form of a quadratic equation is crucial for applying methods like the quadratic formula. The standard form is written as: \[ ax^2 + bx + c = 0 \]. To solve a quadratic equation, it's important to first rewrite it in this form. This involves:
Given \[ 3y^2 - 18y = -27 \], we need to move all terms to one side to get \[ 3y^2 - 18y + 27 = 0 \]. Now, it's in standard form. Here, \( a = 3 \), \( b = -18 \), and \( c = 27 \). Having the equation in standard form simplifies the application of any quadratic solving method, including factoring, completing the square, and using the quadratic formula.
- Combining like terms
- Moving all terms to one side of the equation
- Setting the equation equal to zero
Given \[ 3y^2 - 18y = -27 \], we need to move all terms to one side to get \[ 3y^2 - 18y + 27 = 0 \]. Now, it's in standard form. Here, \( a = 3 \), \( b = -18 \), and \( c = 27 \). Having the equation in standard form simplifies the application of any quadratic solving method, including factoring, completing the square, and using the quadratic formula.
