Problem 50 - Algebra 2

Question

Solve each equation by completing the square. \(x^{2}+1.4 x=1.2\)

Step-by-Step Solution

Answer

The solutions are \(x = 0.6\) and \(x = -2.0\).

Step 1: Move Constant to Other Side

Start by moving the constant term on the left side of the equation to the right side to set up for completing the square:\[ x^2 + 1.4x = 1.2 \]

Step 2: Find the Number to Complete the Square

To complete the square, determine the number that needs to be added to both sides. This number is calculated by taking half of the coefficient of \(x\), which is 1.4, and squaring it:\[ \left(\frac{1.4}{2}\right)^2 = 0.7^2 = 0.49 \]Add 0.49 to both sides of the equation.

Step 3: Add to Both Sides

Add 0.49 to both sides to maintain equality:\[ x^2 + 1.4x + 0.49 = 1.2 + 0.49 \]This simplifies to:\[ x^2 + 1.4x + 0.49 = 1.69 \]

Step 4: Write as a Perfect Square

Now, express the left side of the equation as a square:\[ (x + 0.7)^2 = 1.69 \]

Step 5: Solve for x

Take the square root of both sides to solve for \(x\):\[ x + 0.7 = \pm \sqrt{1.69} \]Simplify:\[ x + 0.7 = \pm 1.3 \]

Step 6: Isolate x

Subtract 0.7 from both sides to solve for \(x\):For the positive root:\[ x = 1.3 - 0.7 = 0.6 \]For the negative root:\[ x = -1.3 - 0.7 = -2.0 \]

Key Concepts

Solving Quadratic Equations

Quadratic equations often appear in the form of \[ ax^2 + bx + c = 0 \]. These equations can be challenging, so we use several methods to find their roots. One such method is completing the square. By doing so, we can rewrite the expression in a simpler form that represents a perfect square trinomial.
When solving quadratic equations by completing the square, we often follow these steps:

• Rearrange the quadratic equation so that the terms involving the variable are on one side.This may involve moving a constant to the other side of the equation. In our example, this meant moving 1.2 to the other side.
• Find the number that should be added to both sides to make a perfect square trinomial. This requires taking half of the linear coefficient, squaring it, and adding it to both sides.
• Express the trinomial as a squared binomial, which allows you to simplify using the square root.
• Finally, solve for the variable by isolating it on one side of the equation.

Understanding how to solve these equations aids in acquiring a foundational skill used across various math and science subjects.

Perfect Square Trinomial

A perfect square trinomial is a notable form of a quadratic expression. It takes the shape of \( (x + d)^2 \), where \( d \) is a constant. Understanding this form aids significantly in handling quadratic expressions.
Creating a perfect square trinomial involves adjusting the coefficients in such a way that the quadratic is effectively a square of a binomial. In our exercise, starting with \( x^2 + 1.4x \), by adding 0.49 to it we form the trinomial \( x^2 + 1.4x + 0.49 \).
This can be rewritten as the perfect square \( (x + 0.7)^2 = 1.69 \) by factoring and ensuring both sides of the equation were adjusted properly.
Being able to recognize and create these forms helps in solving and interpreting quadratic equations,streamlining the solution process.

Quadratic Formula

The quadratic formula is a key technique for solving any quadratic equation, written as \[ ax^2 + bx + c = 0 \]. It provides a direct solution path and is stated as:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \].
This formula is essential, especially when typical methods like factoring aren't straightforward.It covers all possible cases, whether the roots are real or complex.
Though the exercise at hand doesn't require using the quadratic formula, knowing it offers a universal tool for solving quadratics.
Learning how it derives from completing the square gives added insights into its relevance and functionality.Users apply it widely to achieve exact solutions when simpler techniques fall short.