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Magazine Chapter 3: Problem 26
Solve for \(x .\) Be sure to list all possible values of \(x\). $$ 2(x-3)^{2}-7=-4 x+9 $$
Short Answer
Expert verified
The possible values for \(x\) are \(2 + \sqrt{3}\) and \(2 - \sqrt{3}\).
Step by step solution
01
Simplify the Equation
We start with the original equation: \( 2(x-3)^2 - 7 = -4x + 9 \). Our goal is to simplify it by adding or subtracting terms on both sides. Add 7 to both sides to begin grouping terms: \( 2(x-3)^2 = -4x + 16 \).
02
Express as a Quadratic Equation
To form a standard quadratic equation, we'll work on balancing both sides by moving all terms to one side. Add \( 4x \) to both sides to obtain \( 2(x-3)^2 + 4x - 16 = 0 \).
03
Expand the Squared Term
The equation has a squared term \((x-3)^2\). Expand this square: \((x-3)^2 = x^2 - 6x + 9\). Substituting back, we have \(2(x^2 - 6x + 9) + 4x - 16 = 0\).
04
Simplify the Expanded Equation
Distribute the 2 across \(x^2 - 6x + 9\) to obtain \(2x^2 - 12x + 18\). Substituting, the equation becomes \(2x^2 - 12x + 18 + 4x - 16 = 0\). Simplify further by combining like terms, resulting in \(2x^2 - 8x + 2 = 0\).
05
Solve the Quadratic Equation
To solve \(2x^2 - 8x + 2 = 0\), divide the entire equation by 2 to simplify: \(x^2 - 4x + 1 = 0\). Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=-4\), \(c=1\). Calculate the discriminant: \( (-4)^2 - 4(1)(1) = 16 - 4 = 12\).
06
Calculate the Roots
With the discriminant calculated, find the roots: \(x = \frac{4 \pm \sqrt{12}}{2}\). Simplify \(\sqrt{12}\) to \(2\sqrt{3}\), leading to \(x = \frac{4 \pm 2\sqrt{3}}{2}\) which simplifies to \(x = 2 \pm \sqrt{3}\). Thus, the solutions are \(x = 2 + \sqrt{3}\) and \(x = 2 - \sqrt{3}\).
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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 handy tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It allows us to find the values of \( x \) (roots) without needing to factor the equation. Here is the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- The letters \( a \), \( b \), and \( c \) represent the coefficients from the quadratic equation.
- \( \pm \) means there are typically two solutions for \( x \): one involving addition and the other subtraction.
Discriminant
The discriminant is a critical part of the quadratic formula, given by \( b^2 - 4ac \). It helps determine the nature of the roots of a quadratic equation.
- If the discriminant is positive (\( > 0 \)), there are two distinct real roots.
- If it is zero (\( = 0 \)), there is exactly one real root (a repeated root).
- If it is negative (\( < 0 \)), there are no real roots, but two complex roots.
Expanding Binomials
Expanding binomials often appears in quadratic equations. It involves rewriting squared expressions into a standard form. For example, if we have \((x - 3)^2\), we expand it as: \[ (x - 3)^2 = x^2 - 6x + 9 \]
- First, square the first term: \( x^2 \).
- Next, multiply the terms together within the binomial, double it: \( -6x \).
- Lastly, square the second term: \( 9 \).
Simplifying Expressions
Simplifying expressions is crucial in solving equations as it leads to a cleaner and more manageable problem. It involves consolidating and reducing terms to their simplest form. For example, from the equation \(2x^2 - 12x + 18 + 4x - 16 = 0\):
- Combine like terms: \(-12x + 4x = -8x \).
- Simplify constant terms: \(18 - 16 = 2 \).
