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Chapter 3: Problem 14

Use the method of completing the square to solve the quadratic equation. $$x^{2}+4 x-3=0$$

Short Answer

Expert verified
The solutions are \(x = -2 + \sqrt{7}\) and \(x = -2 - \sqrt{7}\).

Step by step solution

01

Move Constant to Other Side

The original equation is \(x^2 + 4x - 3 = 0\). First, move the constant \(-3\) to the right side of the equation: \(x^2 + 4x = 3\).
02

Identify Coefficient and Complete the Square

Take the coefficient of \(x\), which is 4, divide it by 2, and then square it: \(\left(\frac{4}{2}\right)^2 = 4\). Add this squared value (4) to both sides of the equation: \(x^2 + 4x + 4 = 3 + 4\).
03

Write as a Perfect Square

The left side of the equation can now be rewritten as a perfect square trinomial: \((x + 2)^2 = 7\).
04

Solve for x by Taking Square Root

Take the square root of both sides: \(x + 2 = \pm\sqrt{7}\).
05

Isolate x

Solve for \(x\) by subtracting 2 from both sides: \(x = -2 \pm \sqrt{7}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square
Completing the square is a technique used to solve quadratic equations. It transforms a standard quadratic equation into a perfect square trinomial, making it easier to find the roots.
Instead of factorizing directly, we rearrange and modify the equation to make it a complete square.
  • First, ensure the quadratic polynomial is in the form \(ax^2 + bx + c = 0\).
  • Move the constant term \(c\) to the other side, giving you \(ax^2 + bx = -c\).
  • Divide all terms by \(a\) if \(aeq 1\) to simplify.
  • Take the coefficient of \(x\), divide it by 2, and square it. Add this square to both sides of the equation. This step transforms the left side into a perfect square trinomial.
This process not only simplifies solving the equation but also enhances your understanding of quadratic functions.
Perfect Square Trinomial
A perfect square trinomial is an expression of the form \((x + d)^2\), which expands to \(x^2 + 2dx + d^2\). Recognizing and creating perfect square trinomials is crucial in solving quadratics by completing the square.
When completing the square, you are essentially trying to rearrange the quadratic expression into this recognizable form.
  • After adding \(\left(\frac{b}{2}\right)^2\) to both sides, the left-hand side becomes a perfect square trinomial.
  • For example, given \(x^2 + 4x + 4\), this is the result of \((x + 2)^2\).
By converting to this form, you can easily take the square root of both sides to solve for \(x\). This step is vital as it ultimately simplifies the calculation of the equation's solutions.
Quadratic Formula
While completing the square is a highly visual and conceptual approach, the quadratic formula provides a straightforward method to solve any quadratic equation: \(ax^2 + bx + c = 0\).
The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to find the roots of a quadratic equation directly without transforming it first.
  • Identify the coefficients: \(a\), \(b\), and \(c\) from the equation.
  • Substitute them into the formula to find the values of \(x\).
  • \(b^2 - 4ac\) is the discriminant, which indicates the nature of the roots.
The quadratic formula is invaluable when an equation is too complex for factorization or completing the square. It provides a fail-safe solution for any type of quadratic equation.

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Most popular questions from this chapter

Let \(P(x)=x^{3}+p x^{2}+q x+r .\) Two of the zeros of \(P(x)=0\) are 3 and \(1+4 i\). Find the value of \(p, q\) and \(r\)

Let \(f(x)=x^{2}-4 x-1 .\) a) Use the quadratic formula to find the zeros of the function. b) Use the zeros to find the equation for the axis of symmetry of the parabola. c) Find the minimum or maximum value of \(f\)

Sketch the graph of the rational function without the aid of your GDC. On your sketch clearly indicate any \(x\) - or \(y\) -intercepts and any asymptotes (vertical, horizontal or oblique). Use your GDC to verify your sketch. $$h(x)=\frac{x^{2}+2 x}{x-1}$$

Given that \(x=-6\) is a zero of the polynomial \(x^{3}+2 x^{2}-19 x+30\) find all remaining zeros of the polynomial.

Sketch the graph of the rational function without the aid of your GDC. On your sketch clearly indicate any \(x\) - or \(y\) -intercepts and any asymptotes (vertical, horizontal or oblique). Use your GDC to verify your sketch. $$M(x)=\frac{x^{2}+1}{x}$$
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