Solve each equation using completing the square. -2x2 + 4x + 3 = 0

Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

All textbooks

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 45

Chapter 2, Problem 45

Solve each equation using completing the square. -2x² + 4x + 3 = 0

Verified step by step guidance

First, rewrite the equation to isolate the quadratic term with a positive coefficient. Start with the equation \(-2x^2 + 4x + 3 = 0\). Divide every term by \(-2\) to get \(x^2 - 2x - \frac{3}{2} = 0\).

Next, move the constant term to the other side of the equation: \(x^2 - 2x = \frac{3}{2}\).

To complete the square, take half of the coefficient of \(x\), which is \(-2\), divide it by 2 to get \(-1\), and then square it to get \(1\). Add this value to both sides: \(x^2 - 2x + 1 = \frac{3}{2} + 1\).

Rewrite the left side as a perfect square trinomial: \((x - 1)^2 = \frac{3}{2} + 1\). Simplify the right side by expressing \(1\) as \(\frac{2}{2}\) and adding the fractions.

Finally, take the square root of both sides to solve for \(x\): \(x - 1 = \pm \sqrt{\text{right side}}\). Then isolate \(x\) by adding \(1\) to both sides.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves creating a binomial squared expression on one side, making it easier to solve for the variable by taking square roots.

Quadratic Equation Standard Form

A quadratic equation is typically written in the form ax² + bx + c = 0. To complete the square, the equation must be manipulated into this form, often requiring dividing through by the coefficient of x² if it is not 1.

Isolating the Variable

After completing the square, the equation is rewritten so that the variable term is isolated on one side. This allows taking the square root of both sides, leading to two possible solutions due to the ± sign.

Recommended video:

05:28

Equations with Two Variables

Related Practice

Textbook Question

Solve each equation or inequality. |6 - 2x | + 1 = 3

views

Textbook Question

Solve each formula for the specified variable. Assume that the denominator is not 0 if variables appear in the denominator. F = GMm/r², for m (force of gravity)

Textbook Question

Solve each equation. x - √(2x+3) = 0

Textbook Question

Solve each quadratic inequality. Give the solution set in interval notation. 3x²+x≤4

Textbook Question

Solve each equation. x²- √(5x) -1 = 0

views

Textbook Question

Solve each equation or inequality. |8 - 3x| - 3 = -2