Problem 50 - College Algebra

Question

Solve each equation in Exercises \(39-54\) by completing the square. $$ 2 x^{2}-7 x+3=0 $$

Step-by-Step Solution

Answer

The solutions for the equation are \(x=3\) and \(x=0.5\)

Step 1: Rewrite the Equation

First, divide through by the coefficient of \(x^2\), which is 2. The equation then becomes: \(x^2-\frac{7}{2}x+\frac{3}{2}=0\).

Step 2: Completing the Square

Rearrange the equation to isolate the constant on one side. Subtract \(\frac{3}{2}\) from both sides to get:\(x^2-\frac{7}{2}x=-\frac{3}{2}\).Add the square of half the coefficient of x (i.e., \(\left(\frac{-7}{4}\right)^2=\frac{49}{16}\)) to both sides to complete the square:\(x^2-\frac{7}{2}x+\frac{49}{16}=-\frac{3}{2}+\frac{49}{16}\).This simplifies to \( (x-\frac{7}{4})^2 = -\frac{24}{16}+\frac{49}{16} = \frac{25}{16}\).

Step 3: Solve for x

Take the square root of both sides and solve for x:\(x-\frac{7}{4}=\pm\sqrt{\frac{25}{16}}\)This simplifies to: \(x-\frac{7}{4}=\pm\frac{5}{4}\)Finally, solve for x to get: \(x=\frac{7}{4}+\frac{5}{4}=3\) or \(x=\frac{7}{4}-\frac{5}{4}=0.5 \)

Key Concepts

Quadratic Equations

At the heart of algebra, quadratic equations play a fundamental role in various mathematical contexts. A quadratic equation is any equation that can be rearranged in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The solutions to these equations can take on different forms, depending on the discriminant \( b^2 - 4ac \). This determines whether the roots are real, complex, or repeated. Quadratic equations can be recognized by their characteristic parabolic graphs.

• Solving quadratic equations involves finding the values of \( x \) that satisfy the equation.
• The roots of the equation are the points where the parabola intersects the x-axis.
• Common methods for solving these equations include factoring, using the quadratic formula, and completing the square.

Algebraic Methods

Algebraic methods encompass the systematic techniques used to manipulate and solve equations. To solve quadratic equations like \( 2x^2 - 7x + 3 = 0 \), several efficient approaches can be applied. Completing the square is one such algebraic method which repositions the original quadratic into a perfect square trinomial, making it easier to solve.

• The Completing the Square method requires transforming the equation so that the left-hand side becomes a perfect square trinomial.
• It involves isolating the \( x^2 \) and \( x \) terms and then strategically adding a term to both sides to form a complete square.
• This approach simplifies the equation dramatically, creating a straightforward path to finding the roots of the quadratic equation.

Solving Equations

Solving equations is a crucial skill in algebra, blending logic and strategy to find unknown values. For the equation \( x^2 - \frac{7}{2}x + \frac{49}{16} = \frac{25}{16} \), this procedure involves multiple steps:

• First, after transforming the equation to have a perfect square on one side, the task is to take the square root of both sides of the equation.
• This results in two potential solutions because both the positive and negative roots of a square should be considered.
• For the equation given, you find solutions as \( x = 3 \) and \( x = 0.5 \), displaying how a methodical strategy can uncover the values satisfying the original equation.
• The careful manipulation of terms and use of algebraic principles is key to successfully solving equations by this method.